A389406 Expansion of (1/x) * Series_Reversion( x * (1 - x^3 * (1 + x)^2) ).

1, 0, 0, 1, 2, 1, 4, 18, 30, 44, 162, 455, 840, 1960, 5984, 14586, 31920, 84455, 230384, 556094, 1361646, 3641820, 9479340, 23579010, 60700770, 160563285, 414072208, 1059744796, 2775186216, 7288939108, 18903324352, 49237797323, 129596357176, 340168677628, 889621606770

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+k,k) * binomial(2*k,n-3*k).

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

Mathematica

Table[SeriesCoefficient[1/(1-x^3*(1+x)^2)^(n+1), {x, 0, n}]/(n+1), {n, 0, 30}] (* Vincenzo Librandi, Oct 17 2025 *)

Prog

(PARI) my(N=40, x='x+O('x^N)); Vec(serreverse(x*(1-x^3*(1+x)^2))/x)
(Magma) [1/(n+1)*&+[Binomial(n+k, k)*Binomial(2*k, n-3*k): k in [0..Floor(n/3)]]: n in [0..35]]; // Vincenzo Librandi, Oct 17 2025

Crossrefs

Cf. A214372, A389294.

Sequence in context: A267377 A132945 A192494 * A013156 A012925 A012930

Adjacent sequences: A389403 A389404 A389405 * A389407 A389408 A389409

Keyword

nonn

Author

Seiichi Manyama, Oct 03 2025

Status

approved