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

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

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

Cf. A000108, A086616, A162482, A360045.

Sequence in context: A366735 A307733 A045501 * A280208 A088655 A302288

Adjacent sequences: A162478 A162479 A162480 * A162482 A162483 A162484

Keyword

easy,nonn

Author

Paul Barry, Jul 04 2009

Status

approved