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

0,3

Links

Table of n, a(n) for n=0..22.

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.

Example

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

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 *)

Prog

(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)))}

Crossrefs

Cf. A121689, A000108.

Sequence in context: A110576 A074600 A064641 * A307389 A104252 A018977

Adjacent sequences: A183605 A183606 A183607 * A183609 A183610 A183611

Keyword

nonn

Author

Paul D. Hanna, Jan 15 2011

Status

approved