A389632 Expansion of (1/x) * Series_Reversion( x / (1 + x^2 * (1 + x)^2)^2 ).

1, 0, 2, 4, 11, 44, 130, 480, 1717, 6156, 23114, 86020, 326110, 1247220, 4794570, 18594916, 72465228, 283922848, 1117777030, 4417975380, 17529583825, 69787896620, 278695745200, 1116141057196, 4481592026198, 18038043585800, 72762800306980, 294118095053640

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

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

Mathematica

Table[(1/(n+1))*Sum[Binomial[2*(n+1), k]*Binomial[2*k, n-2*k], {k, 0, Floor[n/2]}], {n, 0, 27}] (* Vincenzo Librandi, Nov 10 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+1), k)*Binomial(2*k, n-2*k): k in [0..Floor(n/2)]] : n in [0..30] ]; // Vincenzo Librandi, Nov 10 2025

Crossrefs

Cf. A365129.

Sequence in context: A328437 A107703 A140838 * A114954 A134019 A120259

Adjacent sequences: A389629 A389630 A389631 * A389633 A389634 A389635

Keyword

nonn

Author

Seiichi Manyama, Oct 09 2025

Status

approved