Stirling_1 type
========================================================================================

%S A157386 1,1,1,6,1,18,42,1,144,168,336,1,600,2940,1680,3024,1,4950,33600,35280,18144,
%T A157386 30240,1,26586,336630,717360,444528,211680,332640,1,234528,4870992,11313120,
%U A157386 10329984,5927040,2661120,3991680,

%N A157386 A partition product of Stirling_1 type [parameter k = -6] with biggest-part
           statistic (triangle read by rows).

%C A157386 Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = -6, \Q
           summed over parts with equal biggest part (see the Luschny link). \Q
           Underlying partition triangle is A144356. \Q
           Same partition product with length statistic is A049374. \Q
           Diagonal a(A000217(n)) = rising factorial(6,n-1), A001725(n+4). \Q
           Row sum is A049402.

%F A157386 T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n \Q
           T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that  \Q
           1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!), \Q
           f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-2}(j-n-4).

%Y A157386 Cf. A157385,A157384,A157383,A157382,A126074,A157391,A157392,A157393,A157394,A157395

========================================================================================

%S A157385 1, 1, 1, 5, 1, 15, 30, 1, 105, 120, 210, 1, 425, 1800, 1050, 1680, 1, 3075,
%T A157385 18600, 18900, 10080, 15120, 1, 15855, 174300, 338100, 211680, 105840, 151200,
%U A157385 1, 123515, 2227680, 4865700, 4327680, 2540160, 1209600, 1663200, 1, 757755

%N A157385 A partition product of Stirling_1 type [parameter k = -5] with biggest-part
           statistic (triangle read by rows).

%C A157385 Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = -5, \Q
           summed over parts with equal biggest part (see the Luschny link). \Q
           Underlying partition triangle is A144355. \Q
           Same partition product with length statistic is A049353. \Q
           Diagonal a(A000217(n)) = rising factorial(5,n-1), A001720(n+3). \Q
           Row sum is A049378.

%F A157385 T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n \Q
           T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that  \Q
           1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!), \Q
           f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-2}(j-n-3).

%Y A157385 Cf. A157386,A157384,A157383,A157382,A126074,A157391,A157392,A157393,A157394,A157395

========================================================================================

%S A157384 1, 1, 1, 4, 1, 12, 20, 1, 72, 80, 120, 1, 280, 1000, 600, 840, 1, 1740, 9200,
%T A157384 9000, 5040, 6720, 1, 8484, 79100, 138600, 88200, 47040, 60480, 1, 57232,
%U A157384 874720, 1789200, 1552320, 940800, 483840, 604800, 1, 328752, 9532880

%N A157384 A partition product of Stirling_1 type [parameter k = -4] with biggest-part
           statistic (triangle read by rows).

%C A157384 Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = -4, \Q
           summed over parts with equal biggest part (see the Luschny link). \Q
           Underlying partition triangle is A144354. \Q
           Same partition product with length statistic is A049352. \Q
           Diagonal a(A000217(n)) = rising factorial(4,n-1), A001715(n+2). \Q
           Row sum is A049377.

%F A157384 T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n \Q
           T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that  \Q
           1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!), \Q
           f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-2}(j-n-2).

%Y A157384 Cf. A157386,A157385,A157383,A157382,A126074,A157391,A157392,A157393,A157394,A157395

========================================================================================

%S A157383 1, 1, 1, 3, 1, 9, 12, 1, 45, 48, 60, 1, 165, 480, 300, 360, 1, 855, 3840,
%T A157383 3600, 2160, 2520, 1, 3843, 29400, 46200, 30240, 17640, 20160, 1, 21819,
%U A157383 272832, 520800, 443520, 282240, 161280, 181440

%N A157383 A partition product of Stirling_1 type [parameter k = -3] with biggest-part
           statistic (triangle read by rows).

%C A157383 Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = -3, \Q
           summed over parts with equal biggest part (see the Luschny link). \Q
           Underlying partition triangle is A144353. \Q
           Same partition product with length statistic is A046089. \Q
           Diagonal a(A000217(n)) = rising factorial(3,n-1), A001710(n+1). \Q
           Row sum is A049376.

%F A157383 T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n \Q
           T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that  \Q
           1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!), \Q
           f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-2}(j-n-1).

%Y A157383 Cf. A157386,A157385,A157384,A157382,A126074,A157391,A157392,A157393,A157394,A157395

========================================================================================

%I A126074 !!!!!!!!!!!!!!!!!! Extension only 

%S A126074 1, 1, 1, 1, 1, 3, 2, 1, 9, 8, 6, 1, 25, 40, 30, 24, 1, 75, 200, 180, 144,
%T A126074 120, 1, 231, 980, 1260, 1008, 840, 720, 1, 763, 5152, 8820, 8064, 6720, 5760,
%U A126074 5040, 1, 2619, 28448, 61236, 72576, 60480, 51840, 45360, 40320

%N A126074 Triangle read by rows: T(n,k) is the number of permutations of n elements
           that have the longest cycle length k.

%C A126074 Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = -1, \Q
           summed over parts with equal biggest part (see the Luschny link). \Q
           Underlying partition triangle is A102189. \Q
           Same partition product with length statistic is A008275. \Q
           Diagonal a(A000217(n)) = rising factorial(1,n-1), A000142(n-1) (n > 0). \Q
           Row sum is A000142.

%F A126074 T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n \Q
           T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that  \Q
           1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!), \Q
           f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-2}(j-n+1).

%Y A126074 Cf. A157386,A157385,A157384,A157383,A157382,A157391,A157392,A157393,A157394,A157395

========================================================================================

%S A157391 1, 1, 1, 1, 1, 3, 0, 1, 9, 0, 0, 1, 25, 0, 0, 0, 1, 75, 0, 0, 0, 0, 1, 231,
%T A157391 0, 0, 0, 0, 0, 1, 763, 0, 0, 0, 0, 0, 0, 1, 2619, 0, 0, 0, 0, 0, 0, 0, 1,
%U A157391 9495, 0, 0, 0, 0, 0, 0, 0, 0, 1, 35695, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 140151

%N A157391 A partition product of Stirling_1 type [parameter k = 1] with biggest-part
           statistic (triangle read by rows).

%C A157391 Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = 1, \Q
           summed over parts with equal biggest part (see the Luschny link). \Q
           Underlying partition triangle is A144357. \Q
           Same partition product with length statistic is A049403. \Q
           Diagonal a(A000217(n)) = falling factorial(1,n-1), row in A008279. \Q
           Row sum is A000085.

%F A157391 T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n \Q
           T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that  \Q
           1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!), \Q
           f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-2}(j-n+3).

%Y A157391 Cf. A157386,A157385,A157384,A157383,A157382,A126074,A157392,A157393,A157394,A157395

========================================================================================

%S A157392 1, 1, 1, 2, 1, 6, 2, 1, 24, 8, 0, 1, 80, 60, 0, 0, 1, 330, 320,0, 0, 0, 1,
%T A157392 1302, 2030, 0, 0, 0, 0, 1, 5936, 12432, 0, 0, 0, 0, 0, 1, 26784, 81368, 0, 0,
%U A157392 0, 0, 0, 0, 1, 133650, 545120, 0, 0, 0, 0, 0, 0, 0, 1, 669350, 3825690

%N A157392 A partition product of Stirling_1 type [parameter k = 2] with biggest-part
           statistic (triangle read by rows).

%C A157392 Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = 2, \Q
           summed over parts with equal biggest part (see the Luschny link). \Q
           Underlying partition triangle is A144358. \Q
           Same partition product with length statistic is A049404. \Q
           Diagonal a(A000217(n)) = falling factorial(2,n-1), row in A008279 \Q
           Row sum is A049425.

%F A157392 T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n \Q
           T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that  \Q
           1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!), \Q
           f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = = product_{j=0..n-2}(j-n+4).

%Y A157392 Cf. A157386,A157385,A157384,A157383,A157382,A126074,A157391,A157393,A157394,A157395

========================================================================================

%S A157393 1,1,1,3,1,9,6,1,45,24,6,1,165,240,30,0,1,855,1560,360,0,0,1,3843,12180,3360,
%T A157393 0,0,0,1,21819,96096,30660,0,0,0,0,1,114075,794304,318276,0,0,0,0,0,1,703215,
%U A157393 6850080,3270960,0,0,0,0,0,0,1,4125495,62516520,35053920,0,0,0,0,0,0,0,1

%N A157393 A partition product of Stirling_1 type [parameter k = 3] with biggest-part
           statistic (triangle read by rows).

%C A157393 Partition product prod_{j=0..n-2}(k-n+j+2) and n! at k = 3, \Q
           summed over parts with equal biggest part (see the Luschny link). \Q
           Underlying partition triangle is A144877. \Q
           Same partition product with length statistic is A049410. \Q
           Diagonal a(A000217(n)) = falling factorial(3,n-1), row in A008279 \Q
           Row sum is A049426.

%F A157393 T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n \Q
           T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that  \Q
           1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!), \Q
           f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = = product_{j=0..n-2}(j-n+5).

%Y A157393 Cf. A157386,A157385,A157384,A157383,A157382,A126074,A157391,A157392,A157394,A157395

========================================================================================

%S A157394 1, 1, 1, 4, 1, 12, 12, 1, 72, 48, 24, 1, 280, 600, 120, 24, 1, 1740, 4560,
%T A157394 1800, 144, 0, 1, 8484, 40740, 21000, 2520, 0, 0, 1, 57232, 390432, 223440,
%U A157394 33600, 0, 0, 0, 1, 328752, 3811248, 2845584, 438480, 0, 0, 0, 0, 1, 2389140

%N A157394 A partition product of Stirling_1 type [parameter k = 4] with biggest-part
           statistic (triangle read by rows).

%C A157394 Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = 4, \Q
           summed over parts with equal biggest part (see the Luschny link). \Q
           Underlying partition triangle is A144878. \Q
           Same partition product with length statistic is A049424. \Q
           Diagonal a(A000217(n)) = falling factorial(4,n-1), row in A008279 \Q
           Row sum is A049427.

%F A157394 T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n \Q
           T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that  \Q
           1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!), \Q
           f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = = product_{j=0..n-2}(j-n+6).

%Y A157394 Cf. A157386,A157385,A157384,A157383,A157382,A126074,A157391,A157392,A157393,A157395

========================================================================================

%S A157395 1, 1, 1, 5, 1, 15, 20, 1, 105, 80, 60, 1, 425, 1200, 300, 120, 1, 3075, 10400,
%T A157395 5400, 720, 120, 1, 15855, 102200, 75600, 15120, 840, 0, 1, 123515, 1149120,
%U A157395 907200, 241920, 20160, 0, 0, 1, 757755, 12783680, 13426560, 3719520, 362880

%N A157395 A partition product of Stirling_1 type [parameter k = 5] with biggest-part
           statistic (triangle read by rows).

%C A157395 Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = 5, \Q
           summed over parts with equal biggest part (see the Luschny link). \Q
           Underlying partition triangle is A144879. \Q
           Same partition product with length statistic is A049411. \Q
           Diagonal a(A000217(n)) = falling factorial(5,n-1), row in A008279 \Q
           Row sum is A049431.

%F A157395 T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n \Q
           T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that  \Q
           1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!), \Q
           f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = = product_{j=0..n-2}(j-n+7).

%Y A157395 Cf. A157386,A157385,A157384,A157383,A157382,A126074,A157391,A157392,A157393,A157394

========================================================================================
Stirling2 type
========================================================================================

%S A157396 1, 1, 1, 6, 1, 18, 66, 1, 144, 264, 1056, 1, 600, 4620, 5280, 22176, 1, 4950,
%T A157396 68640, 110880, 133056, 576576, 1, 26586, 639870, 3141600, 3259872, 4036032,
%U A157396 17873856, 1, 234528, 10759056, 69263040, 105557760, 113008896, 142990848,

%N A157396 A partition product of Stirling_2 type [parameter k = -6] with biggest-part
           statistic (triangle read by rows).

%C A157396 Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = -6, \Q
           summed over parts with equal biggest part (see the Luschny link). \Q
           Underlying partition triangle is A134278. \Q
           Same partition product with length statistic is A049385. \Q
           Diagonal a(A000217) = A008548. \Q
           Row sum is A049412.

%F A157396 T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n \Q
           T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that  \Q
           1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!), \Q
           f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-1}(-5*j - 1).

%Y A157396 Cf. A157397,A157398,A157399,A157400,A080510,A157401,A157402,A157403,A157404,A157405

========================================================================================

%S A157397 1, 1, 1, 5, 1, 15, 45, 1, 105, 180, 585, 1, 425, 2700, 2925, 9945, 1, 3075,
%T A157397 34650, 52650, 59670, 208845, 1, 15855, 308700, 1248975, 1253070, 1461915,
%U A157397 5221125, 1, 123515, 4475520, 23689575, 33972120, 35085960, 41769000,

%N A157397 A partition product of Stirling_2 type [parameter k = -5] with biggest-part
           statistic (triangle read by rows).

%C A157397 Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = -5, \Q
           summed over parts with equal biggest part (see the Luschny link). \Q
           Underlying partition triangle is A134273. \Q
           Same partition product with length statistic is A049029. \Q
           Diagonal a(A000217) = A007696. \Q
           Row sum is A049120.

%F A157397 T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n \Q
           T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that  \Q
           1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!), \Q
           f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-1}(-4*j - 1).

%Y A157397 Cf. A157396,A157398,A157399,A157400,A080510,A157401,A157402,A157403,A157404,A157405

========================================================================================

%S A157398 1, 1, 1, 4, 1, 12, 28, 1, 72, 112, 280, 1, 280, 1400, 1400, 3640, 1, 1740,
%T A157398 15120, 21000, 21840, 58240, 1, 8484, 126420, 401800, 382200, 407680, 1106560,
%U A157398 1, 57232, 1538208, 6370000, 8357440, 8153600, 8852480, 24344320, 1,

%N A157398 A partition product of Stirling_2 type [parameter k = -4] with biggest-part
           statistic (triangle read by rows).

%C A157398 Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = -4, \Q
           summed over parts with equal biggest part (see the Luschny link). \Q
           Underlying partition triangle is A134149. \Q
           Same partition product with length statistic is A035469. \Q
           Diagonal a(A000217) = A007559. \Q
           Row sum is A049119.

%F A157398 T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n \Q
           T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that  \Q
           1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!), \Q
           f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-1}(-3*j - 1).

%Y A157398 Cf. A157396,A157397,A157399,A157400,A080510,A157401,A157402,A157403,A157404,A157405

========================================================================================

%S A157399 1, 1, 1, 3, 1, 9, 15, 1, 45, 60, 105, 1, 165, 600, 525, 945, 1, 855, 5250,
%T A157399 6300, 5670, 10395, 1, 3843, 39900, 91875, 79380, 72765, 135135, 1, 21819,
%U A157399 391440, 1164975, 1323000, 1164240, 1081080, 2027025, 1,

%N A157399 A partition product of Stirling_2 type [parameter k = -3] with biggest-part
           statistic (triangle read by rows).

%C A157399 Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = -3, \Q
           summed over parts with equal biggest part (see the Luschny link). \Q
           Underlying partition triangle is A134144. \Q
           Same partition product with length statistic is A035342. \Q
           Diagonal a(A000217) = A001147. \Q
           Row sum is A049118.

%F A157399 T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n \Q
           T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that  \Q
           1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!), \Q
           f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-1}(-2*j - 1).

%Y A157399 Cf. A157396,A157397,A157398,A157400,A080510,A157401,A157402,A157403,A157404,A157405

========================================================================================

%S A157400 1, 1, 1, 2, 1, 6, 6, 1, 24, 24, 24, 1, 80, 180, 120, 120, 1, 330, 1200, 1080,
%T A157400 720, 720, 1, 1302, 7770, 10920, 7560, 5040, 5040, 1, 5936, 57456, 102480,
%U A157400 87360, 60480, 40320, 40320 

%N A157400 A partition product with biggest-part statistic of Stirling_1 type (with
           parameter k = -2) as well as of Stirling_2 type (with parameter k = -2), 
           (triangle read by rows).

%C A157400 Partition product of prod_{j=0..n-1}((k+1)*j - 1) and n! at k = -2, summed \Q
           over parts with equal biggest part (Stirling_2 type) as well as partition \Q
           product of prod_{j=0..n-2}(k-n+j+2) and n! at k = -2 (Stirling_1 type). \Q
           It shares this property with the signless Lah numbers. \Q
           Underlying partition triangle is A130561. \Q
           Same partition product with length statistic is A105278. \Q
           Diagonal a(A000217) = A000142. \Q
           Row sum is A000262.

%F A157400 T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n \Q
           T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that  \Q
           1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!), \Q
           f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-1}(-j-1) \Q
           OR f_n = product_{j=0..n-2}(j-n) since both have the same absolute value n!.

%Y A157400 Cf. A157396,A157397,A157398,A157399,A080510,A157401,A157402,A157403,A157404,A157405
%Y A157400 Cf. A157386,A157385,A157384,A157383,A126074,A157391,A157392,A157393,A157394,A157395

========================================================================================

%I A080510 !!!!!!!!!!!!!!!!!! Extension only 

%N A080510 Triangle read by rows: T(n,k) gives the number of set partitions of {1,..,n}
           with maximum block length k.

%C A080510 Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = -1, \Q
           summed over parts with equal biggest part (see the Luschny link). \Q
           Underlying partition triangle is A036040. \Q
           Same partition product with length statistic is A008277. \Q
           Diagonal a(A000217) = A000012. \Q
           Row sum is A000110.

%F A080510 T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n \Q
           T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that  \Q
           1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!), \Q
           f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-1}(-1) = (-1)^n.

%Y A080510 Cf. A157396,A157397,A157398,A157399,A157400,A157401,A157402,A157403,A157404,A157405

========================================================================================

%S A157401 1, 1, 1, 1, 1, 3, 3, 1, 9, 12, 15, 1, 25, 60, 75, 105, 1, 75, 330, 450, 630,
%T A157401 945, 1, 231, 1680, 3675, 4410, 6615, 10395, 1, 763, 9408, 30975, 41160, 52920,
%U A157401 83160, 135135, 1, 2619, 56952, 233415, 489510, 555660, 748440, 1216215,

%N A157401 A partition product of Stirling_2 type [parameter k = 1] with biggest-part
           statistic (triangle read by rows).

%C A157401 Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = 1, \Q
           summed over parts with equal biggest part (see the Luschny link). \Q
           Underlying partition triangle is A143171. \Q
           Same partition product with length statistic is A001497. \Q
           Diagonal a(A000217) = A001147. \Q
           Row sum is A001515.

%F A157401 T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n \Q
           T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that  \Q
           1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!), \Q
           f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-1}(2*j - 1).

%Y A157401 Cf. A157396,A157397,A157398,A157399,A157400,A080510,A157402,A157403,A157404,A157405

========================================================================================

%S A157402 1, 1, 1, 2, 1, 6, 10, 1, 24, 40, 80, 1, 80, 300, 400, 880, 1, 330, 2400, 3600,
%T A157402 5280, 12320, 1, 1302, 15750, 47600, 55440, 86240, 209440, 1, 5936, 129360,
%U A157402 588000, 837760, 1034880, 1675520, 4188800, 1, 26784, 1146040, 5856480,

%N A157402 A partition product of Stirling_2 type [parameter k = 2] with biggest-part
           statistic (triangle read by rows).

%C A157402 Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = 2, \Q
           summed over parts with equal biggest part (see the Luschny link). \Q
           Underlying partition triangle is A143172. \Q
           Same partition product with length statistic is A004747. \Q
           Diagonal a(A000217) = A008544. \Q
           Row sum is A015735.

%F A157402 T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n \Q
           T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that  \Q
           1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!), \Q
           f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-1}(3*j - 1).

%Y A157402 Cf. A157396,A157397,A157398,A157399,A157400,A080510,A157401,A157403,A157404,A157405

========================================================================================

%S A157403 1, 1, 1, 3, 1, 9, 21, 1, 45, 84, 231, 1, 165, 840, 1155, 3465, 1, 855, 8610,
%T A157403 13860, 20790, 65835, 1, 3843, 64680, 250635, 291060, 460845, 1514205, 1,
%U A157403 21819, 689136, 3969735, 6015240, 7373520, 12113640, 40883535, 1, 114075,

%N A157403 A partition product of Stirling_2 type [parameter k = 3] with biggest-part
           statistic (triangle read by rows).

%C A157403 Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = 3, \Q
           summed over parts with equal biggest part (see the Luschny link). \Q
           Underlying partition triangle is A143173. \Q
           Same partition product with length statistic is A000369. \Q
           Diagonal a(A000217) = A008545 \Q
           Row sum is A016036.

%F A157403 T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n \Q
           T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that  \Q
           1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!), \Q
           f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-1}(4*j - 1).

%Y A157403 Cf. A157396,A157397,A157398,A157399,A157400,A080510,A157401,A157402,A157404,A157405

========================================================================================

%S A157404 1, 1, 1, 4, 1, 12, 36, 1, 72, 144, 504, 1, 280, 1800, 2520, 9576, 1, 1740,
%T A157404 22320, 37800, 57456, 229824, 1, 8484, 182700, 864360, 1005480, 1608768,
%U A157404 6664896, 1, 57232, 2380896, 16546320, 26276544, 32175360, 53319168, 226606464

%N A157404 A partition product of Stirling_2 type [parameter k = 4] with biggest-part
           statistic (triangle read by rows).

%C A157404 Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = 4, \Q
           summed over parts with equal biggest part (see the Luschny link). \Q
           Underlying partition triangle is A144267. \Q
           Same partition product with length statistic is A011801. \Q
           Diagonal a(A000217) = A008546. \Q
           Row sum is A028575.

%F A157404 T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n \Q
           T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that  \Q
           1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!), \Q
           f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-1}(5*j - 1).

%Y A157404 Cf. A157396,A157397,A157398,A157399,A157400,A080510,A157401,A157402,A157403,A157405

========================================================================================

%S A157405 1, 1, 1, 5, 1, 15, 55, 1, 105, 220, 935, 1, 425, 3300, 4675, 21505, 1, 3075,
%T A157405 47850, 84150, 129030, 623645, 1, 15855, 415800, 2323475, 2709630, 4365515,
%U A157405 415800, 2323475, 2709630, 4365515, 21827575, 1, 123515, 6394080, 51934575,

%N A157405 A partition product of Stirling_2 type [parameter k = 5] with biggest-part
           statistic (triangle read by rows).

%C A157405 Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = 5, \Q
           summed over parts with equal biggest part (see the Luschny link). \Q
           Underlying partition triangle is A144268. \Q
           Same partition product with length statistic is A013988. \Q
           Diagonal a(A000217) = A008543. \Q
           Row sum is A028844.

%F A157405 T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n \Q
           T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that  \Q
           1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!), \Q
           f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-1}(6*j - 1).

%Y A157405 Cf. A157396,A157397,A157398,A157399,A157400,A080510,A157401,A157402,A157403,A157404

========================================================================================