A365878 Expansion of (1/x) * Series_Reversion( x*(1+x)^3*(1-x)^4 ).

1, 1, 5, 17, 83, 381, 1939, 9905, 52544, 282315, 1545130, 8552557, 47880020, 270401515, 1539288570, 8821594865, 50860072024, 294774097800, 1716506373521, 10037592274363, 58920231785426, 347051995986538, 2050627029532225, 12151336260368205

Offset

0,3

Links

Table of n, a(n) for n=0..23.

Index entries for reversions of series

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} (-1)^k * binomial(3*n+k+2,k) * binomial(5*n-k+3,n-k).

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

a(n) = (1/(n+1)) * [x^n] 1/( (1+x)^3 * (1-x)^4 )^(n+1). - Seiichi Manyama, Feb 16 2024

Prog

(PARI) a(n) = sum(k=0, n, (-1)^k*binomial(3*n+k+2, k)*binomial(5*n-k+3, n-k))/(n+1);
(SageMath)
def A365878(n):
    h = binomial(5*n + 3, n) * hypergeometric([-n, 3*(n + 1)], [-5 * n - 3], -1) / (n + 1)
    return simplify(h)
print([A365878(n) for n in range(24)])  # Peter Luschny, Sep 21 2023

Crossrefs

Cf. A365752, A365855.

Cf. A365879, A368079.

Cf. A370269.

Sequence in context: A149753 A149754 A149755 * A363163 A002020 A038183

Adjacent sequences: A365875 A365876 A365877 * A365879 A365880 A365881

Keyword

nonn

Author

Seiichi Manyama, Sep 21 2023

Status

approved