A213402 Expansion of exp( Sum_{n>=1} binomial(2*n^2-1, n^2) * x^n/n ).

1, 1, 18, 8121, 75143314, 12641135924840, 36876057665515782388, 1819829483946334379705083199, 1496946629440335290077827171482478552, 20328210229221517747154723779733726877327901902, 4527425732825492268524878479039840102127714838348518156649

Offset: 0

Author

Paul D. Hanna, Jun 10 2012

Keywords

• nonn

Comments

Self-convolution square-root of A201556.

Compare to the g.f. C(x) = 1 + x*C(x)^2 of the Catalan numbers (A000108): C(x) = exp( Sum_{n>=1} binomial(2*n-1,n) * x^n/n ).

Examples

			G.f.: A(x) = 1 + x + 18*x^2 + 8121*x^3 + 75143314*x^4 + 12641135924840*x^5 +...
where
log(A(x)) = x + 35*x^2/2 + 24310*x^3/3 + 300540195*x^4/4 +...+ C(2*n^2-1,n^2)*x^n/n +...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..40

Crossrefs

Cf. A201556, A200002.

Programs

Mathematica

nmax = 10; b = ConstantArray[0, nmax+1]; b[[1]] = 1; Do[b[[n+1]] = 1/n*Sum[Binomial[2*k^2, k^2]/2*b[[n-k+1]], {k, 1, n}], {n, 1, nmax}]; b  (* Vaclav Kotesovec, Mar 06 2014 *)

PARI

{a(n)=polcoeff(exp(sum(m=1, n, binomial(2*m^2, m^2)/2*x^m/m)+x*O(x^n)), n)}

PARI

{a(n)=if(n==0, 1, (1/n)*sum(k=1, n, binomial(2*k^2, k^2)/2*a(n-k)))}
for(n=0,21,print1(a(n),", "))

Formula

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

a(n) ~ 2^(2*n^2-1) / (sqrt(Pi)*n^2). - Vaclav Kotesovec, Mar 06 2014