A389129 Expansion of (1/x) * Series_Reversion( x / (1 + x^3 * (1 + x)^2) ).

1, 0, 0, 1, 2, 1, 3, 14, 24, 30, 95, 275, 495, 962, 2730, 6678, 13440, 30940, 80580, 187986, 415017, 1005252, 2497341, 5833674, 13609123, 33302390, 81139630, 192357165, 461376630, 1129173435, 2739775116, 6588067083, 16008291624, 39166876840, 95262912580, 231427949940, 565907758628

Offset

0,5

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Formula

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

a(n) = (1/(n+1)) * [x^n] (1 + x^3 * (1 + x)^2)^(n+1).

Mathematica

Table[(1/(n+1)) Coefficient[((1+x^3*(1+x)^2))^(n+1), x, n], {n, 0, 31}] (* Vincenzo Librandi, Sep 28 2025 *)

Prog

(PARI) my(N=40, x='x+O('x^N)); Vec(serreverse(x/(1+x^3*(1+x)^2))/x)
(Magma) R<x> := PolynomialRing(Rationals()); [ (1/(n+1))*Coefficient(((1 + x^3 * (1 + x)^2))^(n+1), n) : n in [0..30] ]; // Vincenzo Librandi, Sep 28 2025

Crossrefs

Cf. A365724, A389064.

Sequence in context: A320327 A366591 A007447 * A153189 A362272 A393497

Adjacent sequences: A389126 A389127 A389128 * A389130 A389131 A389132

Keyword

nonn

Author

Seiichi Manyama, Sep 24 2025

Status

approved