A162481 Expansion of (1/(1-x)^3)*c(x/(1-x)^3), c(x) the g.f. of A000108.
1, 4, 14, 54, 235, 1119, 5658, 29800, 161621, 896198, 5056824, 28938519, 167548937, 979653821, 5776252440, 34305807512, 205039491091, 1232333298174, 7443336041318, 45157243590384, 275051410542141, 1681362181696823, 10311616254855422, 63428758470722109
Offset: 0
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..1000
Programs
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Mathematica
a[n_] := Sum[Binomial[n + 2*k + 2, n - k] * CatalanNumber[k], {k, 0, n}]; Array[a, 22, 0] (* Amiram Eldar, Jun 30 2020 *)
Formula
G.f.: 1/((1-x)^3-x-x^2/((1-x)^3-2*x-x^2/((1-x)^3-2*x-x^2/((1-x)^3-2*x-x^2/(1-... (continued fraction);
a(n) = Sum_{k=0..n} C(n+2k+2,n-k)*A000108(k).
Conjecture: (n+1)*a(n) +2*(1-4*n)*a(n-1) +2*(5*n-3)*a(n-2) +4*(2-n)*a(n-3) +(n-3)*a(n-4) = 0. - R. J. Mathar, Dec 11 2011
G.f. A(x) satisfies: A(x) = 1/(1 - x)^3 + x * A(x)^2. - Ilya Gutkovskiy, Jun 30 2020
a(n) = binomial(n+2,2) + Sum_{k=0..n-1} a(k) * a(n-k-1). - Seiichi Manyama, Jan 23 2023
G.f.: (1 - sqrt(1 - 4*x/(1-x)^3))/(2*x). - Vaclav Kotesovec, Jan 24 2023