A382394 a(n) = Sum_{k=0..n} A128899(n,k)^3.

1, 1, 9, 190, 5705, 204876, 8209278, 354331692, 16140234825, 765868074400, 37525317999884, 1886768082651816, 96906387191038334, 5066711735118128200, 268954195756648761900, 14464077426547576156440, 786729115199980286001225, 43219452658242723841261800

Offset: 0

Author

Seiichi Manyama, Mar 24 2025

Keywords

• nonn

Comments

Let b_k(n) = Sum_{j=0..n} A128899(n,j)^k. b_1(n) = binomial(2*n-1,n) = A088218(n) and b_2(n) = A024492(n-1) for n > 0.

Links

• Pedro J. Miana, Hideyuki Ohtsuka, and Natalia Romero, Sums of powers of Catalan triangle numbers, arXiv:1602.04347 [math.NT], 2016.

Crossrefs

Cf. A024492, A088218.

Cf. A001700, A003161, A128899, A183069.

Programs

PARI

a128899(n, k) = binomial(2*n-2, n-k)-binomial(2*n-2, n-k-2);
a(n) = sum(k=0, n, a128899(n, k)^3);

Formula

a(n) = binomial(2*n,n)/2 * A183069(n) for n > 0.

a(n) = A003161(2*n-1) for n > 0.