A389435 Expansion of (1/x) * Series_Reversion( x * (1 - x) / (1 + x / (1 - x)^3) ).

1, 2, 9, 49, 295, 1893, 12691, 87839, 622891, 4502148, 33045504, 245648373, 1845583603, 13992045824, 106907937064, 822394410410, 6364036250875, 49507418751054, 386940716613973, 3037001122486679, 23927242226400558, 189161809728642991, 1500147606392303866

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Formula

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

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

Mathematica

Table[SeriesCoefficient[((1+x/(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/(1-x)^3))/x)
(Magma) [1/(n+1)*&+[Binomial(n+1, k)*Binomial(2*n+2*k, n-k): k in [0..n]]: n in [0..30]]; // Vincenzo Librandi, Oct 17 2025

Crossrefs

Cf. A006013, A366049.

Cf. A389411, A389436.

Sequence in context: A360103 A382454 A390570 * A370345 A000167 A345750

Adjacent sequences: A389432 A389433 A389434 * A389436 A389437 A389438

Keyword

nonn

Author

Seiichi Manyama, Oct 04 2025

Status

approved