A389661 Expansion of (1/x) * Series_Reversion( x * (1 - x^2 * (1 + x))^2 ).

1, 0, 2, 2, 11, 26, 95, 306, 1045, 3696, 12880, 46800, 168084, 618512, 2272424, 8450904, 31529550, 118404884, 446553796, 1691379888, 6431294914, 24535997500, 93911671490, 360453753660, 1387227140364, 5351642372988, 20692323095400, 80173592291344

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(k,n-2*k).

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

Mathematica

Table[SeriesCoefficient[1/(1-x^2*(1+x))^(2*(n+1)), {x, 0, n}]/(n+1), {n, 0, 30}] (* Vincenzo Librandi, Oct 19 2025 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x^2*(1+x))^2)/x)
(Magma) [1/(n+1)*&+[Binomial(2*n+k+1, k)*Binomial(k, n-2*k): k in [0..Floor(n/2)]]: n in [0..35]]; // Vincenzo Librandi, Oct 19 2025

Crossrefs

Cf. A368961.

Sequence in context: A235606 A175202 A187430 * A151365 A244280 A090527

Adjacent sequences: A389658 A389659 A389660 * A389662 A389663 A389664

Keyword

nonn

Author

Seiichi Manyama, Oct 10 2025

Status

approved