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A358624 Triangle read by rows. The coefficients of the Hahn polynomials in ascending order of powers. T(n, k) = n! * [x^k] hypergeom([-x, -n, n + 1], [1, 1], 1).

%I A358624 #8 Nov 28 2022 05:06:00
%S A358624 1,1,2,2,6,6,6,22,30,20,24,100,170,140,70,120,548,1050,1120,630,252,
%T A358624 720,3528,7476,8820,6720,2772,924,5040,26136,59388,78708,64680,37884,
%U A358624 12012,3432,40320,219168,529896,748440,704550,432432,204204,51480,12870
%N A358624 Triangle read by rows. The coefficients of the Hahn polynomials in ascending order of powers. T(n, k) = n! * [x^k] hypergeom([-x, -n, n + 1], [1, 1], 1).
%D A358624 A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable, Springer-Verlag Berlin Heidelberg, 1991.
%F A358624 The general formula for the Hahn polynomials is H(n, x, N, a, b) = (-1)^n*(Pochhammer(N-n, n)*Pochhammer(b+1, n) / n!)*hypergeom([-n, -x, a + b + n + 1], [b + 1, 1 - N], 1). We consider here the case N = a = b = 0.
%e A358624 [0]     1;
%e A358624 [1]     1,      2;
%e A358624 [2]     2,      6,      6;
%e A358624 [3]     6,     22,     30,     20;
%e A358624 [4]    24,    100,    170,    140,     70;
%e A358624 [5]   120,    548,   1050,   1120,    630,    252;
%e A358624 [6]   720,   3528,   7476,   8820,   6720,   2772,    924;
%e A358624 [7]  5040,  26136,  59388,  78708,  64680,  37884,  12012,  3432;
%e A358624 [8] 40320, 219168, 529896, 748440, 704550, 432432, 204204, 51480, 12870;
%p A358624 H := (n, x) -> n!*hypergeom([-x, -n, n + 1], [1, 1], 1):
%p A358624 for n from 0 to 8 do seq(coeff(simplify(H(n, x)), x, k), k = 0..n) od;
%Y A358624 Cf. A000142, A000984, A001564 (row sums), A133942 (alternating row sums).
%K A358624 nonn,tabl
%O A358624 0,3
%A A358624 _Peter Luschny_, Nov 26 2022