A387669 Expansion of (1/x) * Series_Reversion( x * (1 - x^3 * (1 + x)) / (1 + x)^2 ).

1, 2, 5, 15, 53, 210, 888, 3891, 17440, 79552, 368349, 1727804, 8194935, 39236990, 189387236, 920522460, 4501590156, 22132778596, 109342178805, 542506506843, 2702110634696, 13505903382788, 67721914208184, 340565280986575, 1717245556486590, 8680267724171976

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Formula

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

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

Mathematica

Table[SeriesCoefficient[((1+x)^2/(1-x^3*(1+x)))^(n+1), {x, 0, n}]/(n+1), {n, 0, 30}] (* Vincenzo Librandi, Oct 16 2025 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x^3*(1+x))/(1+x)^2)/x)
(Magma) [1/(n+1)*&+[Binomial(n+k, k)*Binomial(2*n+k+2, n-3*k): k in [0..Floor(n/3)]]: n in [0..30]]; // Vincenzo Librandi, Oct 16 2025

Crossrefs

Cf. A369398, A369599, A387671.

Sequence in context: A249892 A352853 A387734 * A006790 A007548 A360052

Adjacent sequences: A387666 A387667 A387668 * A387670 A387671 A387672

Keyword

nonn

Author

Seiichi Manyama, Oct 04 2025

Status

approved