A325137 Triangle T(n, k) = [x^n] (n + k + x)!/(k + x)! for 0 <= k <= n, read by rows.

1, 1, 1, 2, 5, 1, 6, 26, 12, 1, 24, 154, 119, 22, 1, 120, 1044, 1175, 355, 35, 1, 720, 8028, 12154, 5265, 835, 51, 1, 5040, 69264, 133938, 77224, 17360, 1687, 70, 1, 40320, 663696, 1580508, 1155420, 342769, 46816, 3066, 92, 1

Offset: 0

Author

Peter Luschny, Apr 13 2019

Keywords

• nonn
• tabl

Comments

Sister triangle of A307419.

Examples

			Triangle starts:
[0]      1
[1]      1,       1
[2]      2,       5,        1
[3]      6,      26,       12,        1
[4]     24,     154,      119,       22,       1
[5]    120,    1044,     1175,      355,      35,       1
[6]    720,    8028,    12154,     5265,     835,      51,      1
[7]   5040,   69264,   133938,    77224,   17360,    1687,     70,    1
[8]  40320,  663696,  1580508,  1155420,  342769,   46816,   3066,   92,   1
[9] 362880, 6999840, 19978308, 17893196, 6687009, 1197273, 109494, 5154, 117, 1
   A000142, A001705,  A001712,  A001718, A001724, ...
		

Crossrefs

Row sums: A325138.

Columns are: A000142, A001705, A001712, A001718, A001724.

Cf. A307419.

Programs

Maple

T := (n, k) -> add(binomial(j+k, k)*(k+1)^j*abs(Stirling1(n, j+k)), j=0..n-k);
seq(seq(T(n,k), k=0..n), n=0..8);
# Note that for n > 16 Maple fails (at least in some versions) to compute the
# terms properly. Inserting 'simplify' or numerical evaluation might help.
A325137Row := proc(n) local ogf, ser; ogf := (n, k) -> (n+k+x)!/(k+x)!;
ser := (n, k) -> series(ogf(n,k),x,k+2); seq(coeff(ser(n,k),x,k), k=0..n) end: seq(A325137Row(n), n=0..8);

Formula

T(n, k) = Sum_{j=0..n-k} binomial(j+k, k)*|Stirling1(n, j+k)|*(k+1)^j.