A389062 Expansion of (1/x) * Series_Reversion( x / (1 + x^2 * (1 + x)^2) ).

1, 0, 1, 2, 3, 10, 23, 56, 158, 408, 1107, 3080, 8459, 23738, 67041, 189826, 542474, 1556384, 4483886, 12980724, 37702175, 109877494, 321244761, 941739370, 2767960714, 8155029040, 24078735995, 71241354600, 211179908565, 627102094350, 1865273362335, 5556711196680

Offset

0,4

Links

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

Formula

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

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

Mathematica

Table[(1/(n+1)) Coefficient[(1+ x^2* (1+x)^2)^(n+1), x, n], {n, 0, 35}] (* Vincenzo Librandi, Sep 29 2025 *)

Prog

(PARI) my(N=40, x='x+O('x^N)); Vec(serreverse(x/(1+x^2*(1+x)^2))/x)
(Magma) R<x> := PolynomialRing(Rationals()); [ (1/(n+1))*Coefficient((1+ x^2* (1+x)^2)^(n+1), n) : n in [0..30] ]; // Vincenzo Librandi, Sep 29 2025

Crossrefs

Cf. A378426, A387288.

Sequence in context: A089880 A271740 A383503 * A130002 A320812 A162034

Adjacent sequences: A389059 A389060 A389061 * A389063 A389064 A389065

Keyword

nonn

Author

Seiichi Manyama, Sep 22 2025

Status

approved