login
A162481
Expansion of (1/(1-x)^3)*c(x/(1-x)^3), c(x) the g.f. of A000108.
11
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
0,2
LINKS
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
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 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Jul 04 2009
STATUS
approved