A162481 - OEIS

%I #20 Jan 24 2023 03:07:22

%S 1,4,14,54,235,1119,5658,29800,161621,896198,5056824,28938519,

%T 167548937,979653821,5776252440,34305807512,205039491091,

%U 1232333298174,7443336041318,45157243590384,275051410542141,1681362181696823,10311616254855422,63428758470722109

%N Expansion of (1/(1-x)^3)*c(x/(1-x)^3), c(x) the g.f. of A000108.

%H Seiichi Manyama, <a href="/A162481/b162481.txt">Table of n, a(n) for n = 0..1000</a>

%F 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);

%F a(n) = Sum_{k=0..n} C(n+2k+2,n-k)*A000108(k).

%F 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

%F G.f. A(x) satisfies: A(x) = 1/(1 - x)^3 + x * A(x)^2. - _Ilya Gutkovskiy_, Jun 30 2020

%F a(n) = binomial(n+2,2) + Sum_{k=0..n-1} a(k) * a(n-k-1). - _Seiichi Manyama_, Jan 23 2023

%F G.f.: (1 - sqrt(1 - 4*x/(1-x)^3))/(2*x). - _Vaclav Kotesovec_, Jan 24 2023

%t a[n_] := Sum[Binomial[n + 2*k + 2, n - k] * CatalanNumber[k], {k, 0, n}]; Array[a, 22, 0] (* _Amiram Eldar_, Jun 30 2020 *)

%Y Cf. A000108, A086616, A162482, A360045.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Jul 04 2009