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T12

T_{12}(n,k) = \sum_{j=0}^k(-1)^j\binom{n+1}{j}(k-j+1)^n

Triangular form
1  
1 0  
1 1 0  
1 4 1 0  
1 11 11 1 0  
1 26 66 26 1 0  
1 57 302 302 57 1 0
sum als gcd lcm
1 1 1 1
1 1 1 1
2 0 1 1
6 2 4 4
24 0 11 11
120 16 2 858
720 0 1 17214
Linear form (by rows)
Western A173018 1,1,0,1,1,0,1,4
Eastern A123125 1,0,1,0,1,1,0,1
Rectangular form
1 1 1 1 1 1 1
0 1 4 11 26 57 120
0 1 11 66 302 1191 4293
0 1 26 302 2416 15619 88234
0 1 57 1191 15619 156190 1310354
0 1 120 4293 88234 1310354 15724248
0 1 247 14608 455192 9738114 162512286
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Sub Seq 0 1 2 3
Row A000012 A000295 A000460 A000505
Col A000007 A000012 A000295 A000460
RowDiag A180056 A000000 A000000 A000000
ColDiag A180056 A025585 A000000 A000000
Characteristic SUM ALTSUM LCM GCD
A000142 A009006 A180057 A000000
  MIN MAX CENTRAL MIDDLE
A000007 A009006 A180056 A009006

Maple T12 := proc(n,k) local j; add((-1)^j*(1+k-j)^n*binomial(n+1,j),j=0..k) end:

TeX T_{12}(n,k) = \sum_{j=0}^k(-1)^j\binom{n+1}{j}(k-j+1)^n