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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.

%I A250916 #7 Aug 22 2017 12:05:14
%S A250916 1,2,14,152,2236,41512,930904,24474368,738241424,25132379552,
%T A250916 953267419744,39867845243008,1822779782497216,90453927667906688,
%U A250916 4842249786763758464,278167945047964156928,17069371221016503644416,1114374972408995525243392,77126208846034435924819456
%N 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.
%F A250916 E.g.f.: exp( (1-2*x-2*x^2 - sqrt(1-4*x))/(2*x^2) ).
%F A250916 a(n) ~ 2^(2*n+5/2) * n^(n-1) / exp(n-3). - _Vaclav Kotesovec_, Aug 22 2017
%e A250916 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 A250916 such that log(A(x)) = C(x)^2 - 1,
%e A250916 log(A(x)) = 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 + 429*x^6 + 1430*x^7 +...
%e A250916 where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.
%o A250916 (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 A250916 for(n=0, 20, print1(a(n), ", "))
%o A250916 (PARI) {a(n) = n!*polcoeff(exp((1-2*x - sqrt(1-4*x + x^3*O(x^n)))/(2*x^2) - 1), n)}
%o A250916 for(n=0, 20, print1(a(n), ", "))
%Y A250916 Cf. A251568, A250917, A000108.
%K A250916 nonn
%O A250916 0,2
%A A250916 _Paul D. Hanna_, Dec 06 2014