A389439 Expansion of (1/x) * Series_Reversion( x * (1 - x^2 * (1 + x)^3) / (1 + x) ).

1, 1, 2, 8, 32, 127, 535, 2356, 10610, 48632, 226526, 1069173, 5100894, 24559221, 119184145, 582381598, 2862938773, 14149116917, 70259824979, 350373741575, 1753963883717, 8810853069176, 44400530323406, 224395080323116, 1137077676052267, 5776003929614683, 29406591130058104

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Formula

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

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

Mathematica

Table[SeriesCoefficient[((1+x)/(1-x^2*(1+x)^3))^(n+1), {x, 0, n}]/(n+1), {n, 0, 30}] (* Vincenzo Librandi, Oct 18 2025 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x^2*(1+x)^3)/(1+x))/x)
(Magma) [1/(n+1)*&+[Binomial(n+k, k)*Binomial(n+3*k+1, n-2*k): k in [0..Floor(n/2)]]: n in [0..35]]; // Vincenzo Librandi, Oct 18 2025

Crossrefs

Cf. A000108, A181734, A199247.

Sequence in context: A320654 A392979 A389541 * A333579 A274524 A004171

Adjacent sequences: A389436 A389437 A389438 * A389440 A389441 A389442

Keyword

nonn

Author

Seiichi Manyama, Oct 04 2025

Status

approved