A389693 Expansion of (1/x) * Series_Reversion( x * (1 - x^2 / (1 - x)^2)^2 ).

1, 0, 2, 4, 17, 60, 240, 964, 4004, 16900, 72514, 315020, 1383358, 6129928, 27376188, 123094692, 556796140, 2531889312, 11567394600, 53071061120, 244417096785, 1129542027900, 5236442822520, 24345395063964, 113485786553052, 530298129410920, 2483549105251580, 11655428888813128

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

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

Mathematica

Table[(1/(n+1))*Sum[Binomial[2*n+k+1, k]* Binomial[n-1, n-2*k], {k, 0, Floor[n/2]}], {n, 0, 30}] (* Vincenzo Librandi, Oct 31 2025 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x^2/(1-x)^2)^2)/x)
(Magma) [(1/(n+1))*&+[Binomial(2*n+k+1, k) * Binomial(n-1, n-2*k): k in [0..Floor(n/2)]] : n in [0..30] ]; // Vincenzo Librandi, Oct 31 2025

Crossrefs

Cf. A368957, A389694.

Cf. A000957.

Sequence in context: A276557 A266868 A203177 * A096122 A014522 A020035

Adjacent sequences: A389690 A389691 A389692 * A389694 A389695 A389696

Keyword

nonn

Author

Seiichi Manyama, Oct 11 2025

Status

approved