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%I #7 Aug 22 2017 12:05:14
%S 1,2,14,152,2236,41512,930904,24474368,738241424,25132379552,
%T 953267419744,39867845243008,1822779782497216,90453927667906688,
%U 4842249786763758464,278167945047964156928,17069371221016503644416,1114374972408995525243392,77126208846034435924819456
%N 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.
%F E.g.f.: exp( (1-2*x-2*x^2 - sqrt(1-4*x))/(2*x^2) ).
%F a(n) ~ 2^(2*n+5/2) * n^(n-1) / exp(n-3). - _Vaclav Kotesovec_, Aug 22 2017
%e E.g.f.: A(x) = 1 + 2*x + 14*x^2/2! + 152*x^3/3! + 2236*x^4/4! + 41512*x^5/5! +...
%e such that log(A(x)) = C(x)^2 - 1,
%e log(A(x)) = 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 + 429*x^6 + 1430*x^7 +...
%e where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.
%o (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)}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n) = n!*polcoeff(exp((1-2*x - sqrt(1-4*x + x^3*O(x^n)))/(2*x^2) - 1), n)}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A251568, A250917, A000108.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Dec 06 2014
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