A250916 E.g.f.: exp(C(x)^2 - 1) where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers, A000108.

1, 2, 14, 152, 2236, 41512, 930904, 24474368, 738241424, 25132379552, 953267419744, 39867845243008, 1822779782497216, 90453927667906688, 4842249786763758464, 278167945047964156928, 17069371221016503644416, 1114374972408995525243392, 77126208846034435924819456

Offset: 0

Author

Paul D. Hanna, Dec 06 2014

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + 2*x + 14*x^2/2! + 152*x^3/3! + 2236*x^4/4! + 41512*x^5/5! +...
such that log(A(x)) = C(x)^2 - 1,
log(A(x)) = 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 + 429*x^6 + 1430*x^7 +...
where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.
		

Crossrefs

Cf. A251568, A250917, A000108.

Programs

PARI

{a(n)=local(C=1); for(i=0, n, C=1+x*C^2 +x*O(x^n)); n!*polcoeff(exp(C^2-1), n)}
for(n=0, 20, print1(a(n), ", "))

PARI

{a(n) = n!*polcoeff(exp((1-2*x - sqrt(1-4*x + x^3*O(x^n)))/(2*x^2) - 1), n)}
for(n=0, 20, print1(a(n), ", "))

Formula

E.g.f.: exp( (1-2*x-2*x^2 - sqrt(1-4*x))/(2*x^2) ).

a(n) ~ 2^(2*n+5/2) * n^(n-1) / exp(n-3). - Vaclav Kotesovec, Aug 22 2017