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

%I A177938 #23 May 23 2025 01:13:57
%S A177938 1,1,1,6,5,1,60,47,12,1,840,638,179,22,1,15120,11274,3325,485,35,1,
%T A177938 332640,245004,74524,11985,1075,51,1,8648640,6314664,1961470,336049,
%U A177938 34300,2086,70,1,259459200,188204400,59354028,10630508,1182769,83720
%N A177938 Triangle T(n,k) = (-1)^(k+n)*A054655(n,n-k), 0<=k<n, read by rows.
%H A177938 Takao Komatsu, <a href="https://arxiv.org/abs/2505.09357">Some explicit values of a q-multiple zeta function at roots of unity</a>, arXiv:2505.09357 [math.NT], 2025. See p. 10.
%F A177938 Row generating function: Gamma(x+2*n)/Gamma(x+n) = Sum_{k>=0} T(n,k)*x^k.
%F A177938 T(n, k) = n!*(-1)^k*[x^k] hypergeom([-n, -x + n - 1], [-n], 1). - _Peter Luschny_, Mar 22 2022
%e A177938 [0]         1;
%e A177938 [1]         1,         1;
%e A177938 [2]         6,         5,        1;
%e A177938 [3]        60,        47,       12,        1;
%e A177938 [4]       840,       638,      179,       22,       1;
%e A177938 [5]     15120,     11274,     3325,      485,      35,     1;
%e A177938 [6]    332640,    245004,    74524,    11985,    1075,    51,    1;
%e A177938 [7]   8648640,   6314664,  1961470,   336049,   34300,  2086,   70,  1;
%e A177938 [8] 259459200, 188204400, 59354028, 10630508, 1182769, 83720, 3682, 92, 1;
%t A177938 p[x_, n_] = If[n == 0, 1, (n - 1)! / FunctionExpand[Beta[x + n, n]]];
%t A177938 Table[CoefficientList[p[x, n], x], {n, 0, 8}] // Flatten (* rewritten by _Peter Luschny_, Mar 22 2022 *)
%Y A177938 An unsigned, row-reversed variant of A054655.
%Y A177938 Row sums are apparently A001813.
%Y A177938 Cf. A001813, A054655.
%K A177938 nonn,tabl
%O A177938 0,4
%A A177938 _Roger L. Bagula_, May 15 2010