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%I #13 Aug 08 2018 12:11:14
%S 1,1,3,55,2439,181711,19987863,3019344175,597283032279,
%T 149571915236911,46218017081300823,17270884763586798895,
%U 7677911426885078360919,4005536546107407400763311,2423921346754787141028928983,1684444421472099056470715447215,1332493495574767096115773084870359,1190644894731926448479445174157508911,1193491123893325068744832273320725408343
%N E.g.f. A(x) satisfies: 1 = Sum_{n>=0} ( exp(n*x) - A(x) )^n.
%F E.g.f. A(x) satisfies:
%F (1) 1 = Sum_{n>=0} ( exp(n*x) - A(x) )^n.
%F (2) 1 = Sum_{n>=0} exp(n^2*x) / (1 + exp(n*x)*A(x))^(n+1).
%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 55*x^3/3! + 2439*x^4/4! + 181711*x^5/5! + 19987863*x^6/6! + 3019344175*x^7/7! + 597283032279*x^8/8! + 149571915236911*x^9/9! + 46218017081300823*x^10/10! + ...
%e such that
%e 1 = 1 + (exp(x) - A(x)) + (exp(2*x) - A(x))^2 + (exp(3*x) - A(x))^3 + (exp(4*x) - A(x))^4 + (exp(5*x) - A(x))^5 + (exp(6*x) - A(x))^6 + (exp(7*x) - A(x))^7 + ...
%e Also,
%e 1 = 1/(1 + A(x)) + exp(x)/(1 + exp(x)*A(x))^2 + exp(4*x)/(1 + exp(2*x)*A(x))^3 + exp(9*x)/(1 + exp(3*x)*A(x))^4 + exp(16*x)/(1 + exp(4*x)*A(x))^5 + exp(25*x)/(1 + exp(5*x)*A(x))^6 + exp(36*x)/(1 + exp(6*x)*A(x))^7 + ...
%e RELATED SERIES.
%e log(A(x)) = x + 2*x^2/2! + 48*x^3/3! + 2222*x^4/4! + 169080*x^5/5! + 18843302*x^6/6! + 2872307088*x^7/7! + 571992255662*x^8/8! + 143972732107560*x^9/9! + 44668284142577462*x^10/10! + ...
%e The derivative of e.g.f. A(x) equals the ratio of the series:
%e A'(x) = [ Sum_{n>=1} n^2 * exp(n*x) * ( exp(n*x) - A(x) )^(n-1) ] / [ Sum_{n>=1} n * ( exp(n*x) - A(x) )^(n-1) ]. - _Paul D. Hanna_, Aug 06 2018
%o (PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, (exp(m*x +x*O(x^#A)) - Ser(A))^m ) )[#A] ); n!*A[n+1]}
%o for(n=0,20, print1(a(n),", "))
%Y Cf. A303056, A304641, A305134.
%K nonn
%O 0,3
%A _Paul D. Hanna_, May 16 2018
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