A371401 Triangle read by rows: T(n, k) = [x^k] (n*x + 1)*Hypergeometric([-n, -n + 1], [1], x).

1, 1, 1, 1, 4, 4, 1, 9, 21, 9, 1, 16, 66, 76, 16, 1, 25, 160, 340, 205, 25, 1, 36, 330, 1100, 1275, 456, 36, 1, 49, 609, 2905, 5425, 3801, 889, 49, 1, 64, 1036, 6664, 18130, 20776, 9604, 1576, 64, 1, 81, 1656, 13776, 51156, 86436, 65856, 21456, 2601, 81

Offset: 0

Author

Peter Luschny, Mar 22 2024

Keywords

• nonn
• tabl
• easy

Examples

			Triangle starts:
[0] 1;
[1] 1,  1;
[2] 1,  4,    4;
[3] 1,  9,   21,    9;
[4] 1, 16,   66,   76,    16;
[5] 1, 25,  160,  340,   205,    25;
[6] 1, 36,  330, 1100,  1275,   456,   36;
[7] 1, 49,  609, 2905,  5425,  3801,  889,   49;
[8] 1, 64, 1036, 6664, 18130, 20776, 9604, 1576, 64;
		

Crossrefs

Cf. A371400, A097070 (row sums, shifted).

Programs

Maple

P := (n, x) -> (n*x + 1)*hypergeom([-n, -n + 1], [1], x):
T := (n, k) -> coeff(simplify(P(n, x)), x, k):
seq(seq(T(n, k), k = 0..n), n = 0..9);

Formula

Sum_{k=0..n} a(n) = (n + 1)*binomial(2*n - 1, n).