A177938 Triangle T(n,k) = (-1)^(k+n)*A054655(n,n-k), 0<=k

1, 1, 1, 6, 5, 1, 60, 47, 12, 1, 840, 638, 179, 22, 1, 15120, 11274, 3325, 485, 35, 1, 332640, 245004, 74524, 11985, 1075, 51, 1, 8648640, 6314664, 1961470, 336049, 34300, 2086, 70, 1, 259459200, 188204400, 59354028, 10630508, 1182769, 83720

Offset: 0

Author

Roger L. Bagula, May 15 2010

Keywords

• nonn
• tabl

Examples

			[0]         1;
[1]         1,         1;
[2]         6,         5,        1;
[3]        60,        47,       12,        1;
[4]       840,       638,      179,       22,       1;
[5]     15120,     11274,     3325,      485,      35,     1;
[6]    332640,    245004,    74524,    11985,    1075,    51,    1;
[7]   8648640,   6314664,  1961470,   336049,   34300,  2086,   70,  1;
[8] 259459200, 188204400, 59354028, 10630508, 1182769, 83720, 3682, 92, 1;
		

Links

• Takao Komatsu, Some explicit values of a q-multiple zeta function at roots of unity, arXiv:2505.09357 [math.NT], 2025. See p. 10.

Crossrefs

An unsigned, row-reversed variant of A054655.

Row sums are apparently A001813.

Cf. A001813, A054655.

Programs

Mathematica

p[x_, n_] = If[n == 0, 1, (n - 1)! / FunctionExpand[Beta[x + n, n]]];
Table[CoefficientList[p[x, n], x], {n, 0, 8}] // Flatten (* rewritten by Peter Luschny, Mar 22 2022 *)

Formula

Row generating function: Gamma(x+2*n)/Gamma(x+n) = Sum_{k>=0} T(n,k)*x^k.

T(n, k) = n!*(-1)^k*[x^k] hypergeom([-n, -x + n - 1], [-n], 1). - Peter Luschny, Mar 22 2022