A382403 a(n) = Sum_{k=0..n} A039599(n,k)^3.

1, 2, 36, 980, 33040, 1268568, 53105976, 2364239592, 110206067400, 5323547715200, 264576141331216, 13458185494436592, 697931136204820336, 36789784967375728400, 1966572261077797609200, 106400946932857148590800, 5817987630644593688220600, 321105713814359742307398480

Offset: 0

Author

Seiichi Manyama, Mar 24 2025

Keywords

• nonn

Comments

Let b_k(n) = Sum_{j=0..n} A039599(n,j)^k. b_1(n) = binomial(2*n,n) = A000984(n) and b_2(n) = binomial(4*n,2*n)/(2*n+1) = A048990(n).

Links

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

Crossrefs

Cf. A000984, A048990.

Cf. A039599, A112029.

Programs

PARI

a039599(n, k) = (2*k+1)/(n+k+1)*binomial(2*n, n-k);
a(n) = sum(k=0, n, a039599(n, k)^3);

Formula

a(n) = binomial(2*n,n) * (4 * binomial(2*n,n)^2 - 3 * A112029(n)).