A158266 G.f.: A(x) = exp( Sum_{n>=1} C(2n-1,n)^2*x^n/n ).

1, 1, 5, 38, 352, 3659, 41012, 484739, 5959417, 75523708, 980470867, 12980840984, 174675568464, 2382923659387, 32890264803521, 458576476085929, 6450351908991558, 91437202854436755, 1305115286958337403

Offset: 0

Author

Paul D. Hanna, Apr 09 2009

Keywords

• nonn
• easy

Comments

Compare g.f. to Catalan(x) = exp( Sum_{n>=1} C(2n-1,n)*x^n/n ), where C(2n-1,n) = A001700(n-1) and Catalan(x) is the g.f. of A000108(n) = C(2n,n)/(n+1).

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 38*x^3 + 352*x^4 + 3659*x^5 + 41012*x^6 +...
log(A(x)) = x + 3^2*x^2/2 + 10^2*x^3/3 + 35^2*x^4/4 + 126^2*x^5/5 +...
log(C(x)) = x + 3*x^2/2 + 10*x^3/3 + 35*x^4/4 + 126*x^5/5 +...
C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 +... (g.f. of A000108).
		

Crossrefs

Cf. A001700, A000108, A224734.

Programs

PARI

{a(n)=polcoeff(exp(sum(m=1,n,binomial(2*m-1,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-1,k)^2*a(n-k)))}

Formula

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

A(x) = exp( Sum_{n >= 1} 1/4*C(2*n,n)^2*x^n/n ). A(x)^4 is the o.g.f. for A224734. - Peter Bala, Jun 04 2015