A389345 Expansion of (1/x) * Series_Reversion( x * (1 - x^2 / (1 - x)^3) ).

1, 0, 1, 3, 9, 31, 111, 408, 1538, 5910, 23058, 91107, 363828, 1466097, 5953987, 24344088, 100129287, 414013081, 1719903186, 7174984584, 30045945152, 126253865664, 532186707684, 2249715451798, 9535278653280, 40512725620275, 172513099806756, 736129520341155

Offset

0,4

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+k-1,n-2*k).

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

Mathematica

Table[SeriesCoefficient[1/(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))/x)
(Magma) [1/(n+1)*&+[Binomial(n+k, k)*Binomial(n+k-1, n-2*k): k in [0..Floor(n/2)]]: n in [0..35]]; // Vincenzo Librandi, Oct 18 2025

Crossrefs

Cf. A000957, A046736, A389346, A389347.

Cf. A389246.

Sequence in context: A151033 A047012 A120305 * A049170 A049174 A209335

Adjacent sequences: A389342 A389343 A389344 * A389346 A389347 A389348

Keyword

nonn

Author

Seiichi Manyama, Oct 01 2025

Status

approved