A389436 Expansion of (1/x) * Series_Reversion( x * (1 - x) / (1 + x^2 / (1 - x)^3) ).

1, 1, 3, 12, 52, 237, 1125, 5511, 27656, 141440, 734498, 3862630, 20529093, 110094753, 595020077, 3237635901, 17721430460, 97509823152, 539050376936, 2992488640016, 16675518651355, 93243031100736, 523010796873188, 2942024211480109, 16592868304255459, 93809462135031897

Offset

0,3

Links

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

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(n+1,k) * binomial(2*n+k,n-2*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 17 2025 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)/(1+x^2/(1-x)^3))/x)
(Magma) [1/(n+1)*&+[Binomial(n+1, k)*Binomial(2*n+k, n-2*k): k in [0..Floor(n/2)]]: n in [0..30]]; // Vincenzo Librandi, Oct 17 2025

Crossrefs

Cf. A052709, A109081, A389410.

Cf. A389411, A389435.

Sequence in context: A151196 A363090 A010736 * A151197 A007198 A348479

Adjacent sequences: A389433 A389434 A389435 * A389437 A389438 A389439

Keyword

nonn

Author

Seiichi Manyama, Oct 04 2025

Status

approved