OFFSET
0,3
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 15 2011
STATUS
approved