A389246 Expansion of (1/x) * Series_Reversion( x / (1 + x^2 / (1 - x)^3) ).

1, 0, 1, 3, 8, 25, 83, 280, 966, 3402, 12162, 44011, 160941, 593814, 2207816, 8263724, 31112838, 117750432, 447713309, 1709407257, 6551228310, 25192992792, 97181826623, 375944600108, 1458127244445, 5669058115560, 22089785423695, 86251564121931, 337422887622018

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

a(n) = (1/(n+1)) * [x^n] (1 + x^2 / (1 - x)^3)^(n+1).

D-finite with recurrence -3*(3*n+2)*(3*n+1) *(5329638178135*n -6740091898537)*(n+1)*a(n) +(957297182797754*n^4 -2683976978115556*n^3 +2789459719303369*n^2 -911914488473519*n +40440551391222) *a(n-1) +(-2136238900546091*n^4 +15699572589884723*n^3 -39252401421363832*n^2 +39901025026117972*n -14020651307189502) *a(n-2) +6*(574225490711666*n^4 -4669351681023436*n^3 +14992495899086774*n^2 -21603111733355204*n +10995922036219955) *a(n-3) -(n-4) *(4457503378649747*n^3 -27859307256243307*n^2 +51102685535867403*n -25763046371721693) *a(n-4) +(n-4) *(n-5) *(4482834765302906*n^2 -23916955442793690*n +21414530310542049) *a(n-5) -31*(n-4) *(n-5) *(n-6) *(15703570527019*n -18045386290574) *a(n-6)=0. - R. J. Mathar, Sep 29 2025

Mathematica

Table[Coefficient[(1+x)^(n-1) (1+x^2+x^3)^(n+1), x, n]/(n+1), {n, 0, 25}] (* Vincenzo Librandi, Oct 01 2025 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1+x^2/(1-x)^3))/x)
(Magma) a := [ n eq 0 select 1 else (n eq 1 select 0 else &+[Binomial(n+1, k)*Binomial(n+k-1, 3*k-1) : k in [1..Floor(n/2)]]/(n+1)) : n in [0..35] ]; a; // Vincenzo Librandi, Oct 01 2025

Crossrefs

Cf. A161797, A389250.

Sequence in context: A258466 A216640 A148794 * A143330 A148795 A148796

Adjacent sequences: A389243 A389244 A389245 * A389247 A389248 A389249

Keyword

nonn

Author

Seiichi Manyama, Sep 26 2025

Status

approved