A389196 G.f. A(x) satisfies A(x) = 1 + x/(1-x^2) * A(x)^3.

1, 1, 3, 13, 61, 310, 1657, 9190, 52393, 305155, 1807922, 10861507, 66013637, 405162518, 2507635730, 15633283370, 98082119253, 618809792699, 3923563925805, 24988037929779, 159778209864528, 1025347017029433, 6601594402837783, 42631140337468305

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k-1,k) * A001764(n-2*k).

Mathematica

Table[Sum[Binomial[n-k-1, k]*Binomial[3*(n-2*k), n-2*k]/(2*(n-2*k)+1), {k, 0, Floor[n/2]}], {n, 0, 25}] (* Vincenzo Librandi, Nov 14 2025 *)
terms = 24; A[_] = 0; Do[A[x_] =1 + x /(1-x^2)*A[x]^3 + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Nov 16 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\2, binomial(n-k-1, k)*binomial(3*(n-2*k), n-2*k)/(2*(n-2*k)+1));
(Magma) [&+[Binomial(n-k-1, k)*Binomial(3*(n-2*k), n-2*k)/(2*(n-2*k)+1): k in [0..Floor(n/2)]] : n in [0..30] ]; // Vincenzo Librandi, Nov 14 2025

Crossrefs

Cf. A307678, A389285.

Cf. A001764, A085139.

Sequence in context: A200215 A074548 A367059 * A243014 A258799 A375651

Adjacent sequences: A389193 A389194 A389195 * A389197 A389198 A389199

Keyword

nonn,easy

Author

Seiichi Manyama, Oct 26 2025

Status

approved