A389249 Expansion of (1/x) * Series_Reversion( x / (1 + x^3 / (1 - x)^2) ).

1, 0, 0, 1, 2, 3, 7, 19, 46, 109, 273, 702, 1801, 4639, 12087, 31756, 83822, 222273, 592345, 1585434, 4258923, 11478790, 31034826, 84147522, 228748429, 623321433, 1702273077, 4658431177, 12772604283, 35082646584, 96522410620, 265975158449, 733988636766

Offset

0,5

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+1,k) * binomial(n-k-1,n-3*k).

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

Mathematica

a[n_]:=SeriesCoefficient[(1+x^3/(1-x)^2)^(n+1), {x, 0, n}]/(n+1); Table[a[n], {n, 0, 25}] (* Vincenzo Librandi, Oct 01 2025 *)

Prog

(PARI) my(N=40, x='x+O('x^N)); Vec(serreverse(x/(1+x^3/(1-x)^2))/x)
(Magma) a := [ n eq 0 select 1 else &+[Binomial(n+1, k) * Binomial(n-k-1, n-3*k) : k in [0..Floor(n/3)] ] div (n+1): n in [0..30] ]; a; // Vincenzo Librandi, Oct 01 2025

Crossrefs

Cf. A114997, A389250, A389251, A389252.

Cf. A109081, A389245.

Sequence in context: A091410 A069051 A229290 * A246033 A122724 A256758

Adjacent sequences: A389246 A389247 A389248 * A389250 A389251 A389252

Keyword

nonn

Author

Seiichi Manyama, Sep 26 2025

Status

approved