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).

1, 1, 2, 2, 6, 6, 6, 22, 30, 20, 24, 100, 170, 140, 70, 120, 548, 1050, 1120, 630, 252, 720, 3528, 7476, 8820, 6720, 2772, 924, 5040, 26136, 59388, 78708, 64680, 37884, 12012, 3432, 40320, 219168, 529896, 748440, 704550, 432432, 204204, 51480, 12870

Offset: 0

Author

Peter Luschny, Nov 26 2022

Keywords

• nonn
• tabl

Examples

			[0]     1;
[1]     1,      2;
[2]     2,      6,      6;
[3]     6,     22,     30,     20;
[4]    24,    100,    170,    140,     70;
[5]   120,    548,   1050,   1120,    630,    252;
[6]   720,   3528,   7476,   8820,   6720,   2772,    924;
[7]  5040,  26136,  59388,  78708,  64680,  37884,  12012,  3432;
[8] 40320, 219168, 529896, 748440, 704550, 432432, 204204, 51480, 12870;
		

References

• A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable, Springer-Verlag Berlin Heidelberg, 1991.

Crossrefs

Cf. A000142, A000984, A001564 (row sums), A133942 (alternating row sums).

Programs

Maple

H := (n, x) -> n!*hypergeom([-x, -n, n + 1], [1, 1], 1):
for n from 0 to 8 do seq(coeff(simplify(H(n, x)), x, k), k = 0..n) od;

Formula

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.