A365751 Expansion of (1/x) * Series_Reversion( x*(1+x)*(1-x)^3 ).

1, 2, 8, 38, 201, 1134, 6688, 40734, 254237, 1617572, 10452416, 68408626, 452530659, 3020870352, 20324167488, 137672551630, 938154745773, 6426806842566, 44234352581896, 305743015718028, 2121318029754770, 14769052147618740, 103148538125870880

Offset

0,2

Links

Table of n, a(n) for n=0..22.

Index entries for reversions of series

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} (-1)^k * binomial(n+k,k) * binomial(4*n-k+2,n-k).

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

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

Prog

(PARI) a(n) = sum(k=0, n, (-1)^k*binomial(n+k, k)*binomial(4*n-k+2, n-k))/(n+1);
(SageMath)
def A365751(n):
    h = binomial(4*n + 2, n) * hypergeometric([-n, n + 1], [-4 * n - 2], -1) / (n + 1)
    return simplify(h)
print([A365751(n) for n in range(23)])  # Peter Luschny, Sep 20 2023

Crossrefs

Cf. A063020, A365752, A365753.

Cf. A352373.

Sequence in context: A266797 A369208 A234939 * A192784 A108246 A020031

Adjacent sequences: A365748 A365749 A365750 * A365752 A365753 A365754

Keyword

nonn

Author

Seiichi Manyama, Sep 18 2023

Status

approved