A389245 Expansion of (1/x) * Series_Reversion( x / (1 + x^2 / (1 - x)^2) ).

1, 0, 1, 2, 5, 14, 40, 118, 357, 1100, 3441, 10900, 34892, 112696, 366808, 1201942, 3961765, 13126872, 43697395, 146070954, 490127325, 1650206070, 5573390956, 18877202872, 64104934380, 218218591344, 744490926700, 2545212141056, 8718112068552, 29915631822160

Offset

0,4

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

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

Mathematica

a[n_]:=SeriesCoefficient[(1+x^2/(1-x)^2)^(n+1), {x, 0, n}]/(n+1); Table[a[n], {n, 0, 25}] (* Vincenzo Librandi, Oct 01 2025 *)

Prog

(PARI) my(N=40, x='x+O('x^N)); Vec(serreverse(x/(1+x^2/(1-x)^2))/x)
(Magma) a := [&+[Binomial(n+1, k) * Binomial(n-1, n-2*k) : k in [0..Floor(n/2)] ] div (n+1) : n in [0..30] ]; a; // Vincenzo Librandi, Oct 01 2025

Crossrefs

Cf. A109081, A389249.

Cf. A321199.

Sequence in context: A075496 A114177 A349413 * A200438 A363933 A103140

Adjacent sequences: A389242 A389243 A389244 * A389246 A389247 A389248

Keyword

nonn

Author

Seiichi Manyama, Sep 26 2025

Status

approved