A201556 G.f.: exp( Sum_{n>=1} C(2*n^2,n^2) * x^n/n ).

1, 2, 37, 16278, 150303194, 25282422428664, 73752140616074524401, 3639659041645240391812731402, 2993893262520330875797362908273443346, 40656420461436928818704580402413441308206341488, 9054851465691640957562090101797213977192016103053025996396

Offset: 0

Author

Paul D. Hanna, Dec 02 2011

Keywords

• nonn

Comments

Self-convolution of A213402.

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

Examples

			G.f.: A(x) = 1 + 2*x + 37*x^2 + 16278*x^3 + 150303194*x^4 +...
where
log(A(x)) = 2*x + 70*x^2/2 + 48620*x^3/3 + 601080390*x^4/4 +...+ C(2*n^2,n^2)*x^n/n +...
		

Links

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

Crossrefs

Cf. A213402, A201555, A000984, A200002, A213409.

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]*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)*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)*a(n-k)))}

Formula

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

a(n) ~ 4^(n^2) / (sqrt(Pi)*n^2). - Vaclav Kotesovec, Mar 06 2014