A183608 G.f.: A(x) = Sum_{n>=0} x^n * C(x)^(n^2), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

1, 1, 2, 7, 29, 133, 658, 3471, 19400, 114417, 709815, 4619048, 31446579, 223419752, 1652599036, 12698380493, 101151995810, 833740791381, 7098646227614, 62335051895044, 563749889969108, 5244173616702347, 50117689766439784

Offset: 0

Author

Paul D. Hanna, Jan 15 2011

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 29*x^4 + 133*x^5 + 658*x^6 +...
		

Crossrefs

Cf. A121689, A000108.

Programs

Mathematica

Flatten[{1,Table[Sum[Binomial[(n-k)^2+2*k, k] * (n-k)^2/((n-k)^2 + 2*k),{k,0,n-1}],{n,1,20}]}] (* Vaclav Kotesovec, Mar 06 2014 *)

PARI

{a(n)=if(n<0,0,0^n+sum(k=0, n-1, binomial((n-k)^2+2*k, k)*(n-k)^2/((n-k)^2+2*k)))}

Formula

a(n) = Sum_{k=0..n-1} binomial((n-k)^2+2k, k) * (n-k)^2/((n-k)^2 + 2k) for n>0 with a(0)=1.

G.f.: A(x) = Sum_{n>=0} x^n*C(x)^n*Product_{k=1..n} (1-x*C(x)^(4*k-3))/(1-x*C(x)^(4*k-1)) where C(x) = 1 + x*C(x)^2.

Let q = C(x) = 1 + x*C(x)^2, then g.f. A(x) equals the continued fraction:

A(x) = 1/(1- q*x/(1- q*(q^2-1)*x/(1- q^5*x/(1- q^3*(q^4-1)*x/(1- q^9*x/(1- q^5*(q^6-1)*x/(1- q^13*x/(1- q^7*(q^8-1)*x/(1- ...)))))))))

due to an identity of a partial elliptic theta function.

G.f.: A(x) = 1 + x*C(x)* G( x*C(x)^2 ), where G(x) = Sum_{k>=0} x^k*(1+x)^(k^2) is the g.f. of A121689.