A365880 Expansion of (1/x) * Series_Reversion( x*(1+x)^4*(1-x)^5 ).

1, 1, 6, 21, 116, 566, 3176, 17501, 101391, 590756, 3519782, 21163038, 128845344, 790810400, 4894134376, 30486741869, 191068074202, 1203710067455, 7619193325238, 48430121151156, 309011352878208, 1978450305442086, 12706836843595840

Offset: 0

Author

Seiichi Manyama, Sep 21 2023

Keywords

• nonn

Links

• Index entries for reversions of series

Crossrefs

Cf. A365753, A365856, A365879.

Programs

PARI

a(n) = sum(k=0, n, (-1)^k*binomial(4*n+k+3, k)*binomial(6*n-k+4, n-k))/(n+1);

SageMath

def A365880(n):
    h = binomial(6*n + 4, n) * hypergeometric([-n, 4*(n + 1)], [-6 * n - 4], -1) / (n + 1)
    return simplify(h)
print([A365880(n) for n in range(23)])  # Peter Luschny, Sep 21 2023

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} (-1)^k * binomial(4*n+k+3,k) * binomial(6*n-k+4,n-k).

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(4*n+k+3,k) * binomial(2*n-2*k,n-2*k). - Seiichi Manyama, Jan 18 2024

a(n) = (1/(n+1)) * [x^n] 1/( (1+x)^4 * (1-x)^5 )^(n+1). - Seiichi Manyama, Feb 16 2024