A387667 Expansion of (1/x) * Series_Reversion( x * (1 - x^2 * (1 + x)) / (1 + x)^2 ).

1, 2, 6, 23, 100, 467, 2286, 11575, 60124, 318591, 1715424, 9358698, 51621088, 287397239, 1612913922, 9114971598, 51825355386, 296254212732, 1701630535526, 9815849332844, 56841818973890, 330314190834573, 1925612052550394, 11258317334229668, 65998932587344764

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Formula

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

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

Mathematica

Table[SeriesCoefficient[((1+x)^2/(1-x^2*(1+x)))^(n+1), {x, 0, n}]/(n+1), {n, 0, 30}] (* Vincenzo Librandi, Oct 17 2025 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x^2*(1+x))/(1+x)^2)/x)
(Magma) [1/(n+1)*&+[Binomial(n+k, k)*Binomial(2*n+k+2, n-2*k): k in [0..Floor(n/2)]]: n in [0..30]]; // Vincenzo Librandi, Oct 17 2025

Crossrefs

Cf. A006318, A274378, A387668.

Sequence in context: A133656 A078487 A376317 * A193038 A213090 A376388

Adjacent sequences: A387664 A387665 A387666 * A387668 A387669 A387670

Keyword

nonn

Author

Seiichi Manyama, Oct 04 2025

Status

approved