A365854 Expansion of (1/x) * Series_Reversion( x*(1+x)^2*(1-x)^3 ).

1, 1, 4, 13, 55, 232, 1052, 4869, 23206, 112519, 554560, 2767336, 13959941, 71060356, 364569352, 1883143669, 9785481498, 51118097686, 268294595396, 1414106565611, 7481787454031, 39721596068000, 211549545257760, 1129912319370600, 6050931114958080

Offset

0,3

Links

Table of n, a(n) for n=0..24.

Index entries for reversions of series

Formula

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

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

a(n) = (1/(n+1)) * [x^n] 1/( (1+x)^2 * (1-x)^3 )^(n+1). - Seiichi Manyama, Feb 16 2024

Prog

(PARI) a(n) = sum(k=0, n, (-1)^k*binomial(2*n+k+1, k)*binomial(4*n-k+2, n-k))/(n+1);
(SageMath)
def A365854(n):
    h = binomial(2*(2*n + 1), n) * hypergeometric([-n, 2*(n + 1)], [-2*(2*n + 1)], -1) / (n + 1)
    return simplify(h)
print([A365854(n) for n in range(25)])  # Peter Luschny, Sep 20 2023

Crossrefs

Cf. A365855, A365856.

Cf. A063020, A365878, A365880.

Cf. A365751, A370103.

Sequence in context: A149471 A149472 A149473 * A188205 A149474 A149475

Adjacent sequences: A365851 A365852 A365853 * A365855 A365856 A365857

Keyword

nonn

Author

Seiichi Manyama, Sep 20 2023

Status

approved