A388916 Expansion of (1/(3*x)) * Series_Reversion( x / ((1+x)^2 * (x+3*(1+x)^2)) ).

1, 13, 229, 4654, 102766, 2395588, 58018051, 1445559583, 36814892431, 954069925159, 25078529906554, 667026049270594, 17918499388172959, 485458570769113201, 13249366192918472773, 363936845430399270718, 10053396678028761441193, 279112559195478744158917

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} 3^(n-k) * binomial(n+1,k) * binomial(4*n-2*k+4,n-k).

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

Mathematica

Table[(1/(3*(n+1))) Coefficient[((1+x)^2*(x+3*(1+x)^2))^(n+1), x, n], {n, 0, 35}] (* Vincenzo Librandi, Sep 29 2025 *)

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serreverse(x/((1+x)^2*(x+3*(1+x)^2)))/(3*x))
(Magma) R<x> := PolynomialRing(Rationals()); [ (1/(3*(n+1)))*Coefficient(((1+x)^2 * (x+3*(1+x)^2))^(n+1), n) : n in [0..30] ]; // Vincenzo Librandi, Sep 29 2025

Crossrefs

Cf. A364792, A388913.

Sequence in context: A223548 A201706 A176722 * A083081 A388049 A083306

Adjacent sequences: A388913 A388914 A388915 * A388917 A388918 A388919

Keyword

nonn

Author

Seiichi Manyama, Sep 21 2025

Status

approved