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A304641 E.g.f. A(x) satisfies: 1 = Sum_{n>=0} ( exp((n+1)*x) - A(x) )^n.

%I A304641 #7 May 16 2018 14:42:54
%S A304641 1,2,6,74,3078,228842,25277286,3837501194,762731347398,
%T A304641 191798593122602,59475206565622566,22290155840476400714,
%U A304641 9933314218291366691718,5192540728710234994272362,3147427468437058629798524646,2190237887318737512524514442634,1734606000858253287464231519860038,1551466530739217915273113571521758122,1556475858078242120174483544923467343526
%N A304641 E.g.f. A(x) satisfies: 1 = Sum_{n>=0} ( exp((n+1)*x) - A(x) )^n.
%F A304641 G.f. A(x) satisfies:
%F A304641 (1) 1 = Sum_{n>=0} ( exp((n+1)*x) - A(x) )^n.
%F A304641 (2) 1 = Sum_{n>=0} exp(n*(n+1)*x) / (1 + exp(n*x)*A(x))^(n+1).
%e A304641 E.g.f.: A(x) = 1 + 2*x + 6*x^2/2! + 74*x^3/3! + 3078*x^4/4! + 228842*x^5/5! + 25277286*x^6/6! + 3837501194*x^7/7! + 762731347398*x^8/8! + 191798593122602*x^9/9! + 59475206565622566*x^10/10! + ...
%e A304641 such that
%e A304641 1 = 1  +  (exp(2*x) - A(x))  +  (exp(3*x) - A(x))^2  +  (exp(4*x) - A(x))^3  +  (exp(5*x) - A(x))^4  +  (exp(6*x) - A(x))^5  +  (exp(7*x) - A(x))^6  +  (exp(8*x) - A(x))^7 + ...
%e A304641 Also,
%e A304641 1 = 1/(1 + A(x))  +  exp(2*x)/(1 + exp(x)*A(x))^2  +  exp(6*x)/(1 + exp(2*x)*A(x))^3  +  exp(12*x)/(1 + exp(3*x)*A(x))^4  +  exp(20*x)/(1 + exp(4*x)*A(x))^5  +  exp(30*x)/(1 + exp(5*x)*A(x))^6  +  exp(42*x)/(1 + exp(6*x)*A(x))^7 + ...
%e A304641 RELATED SERIES.
%e A304641 log(A(x)) = 2*x + 2*x^2/2! + 54*x^3/3! + 2570*x^4/4! + 199590*x^5/5! + 22598762*x^6/6! + 3488755494*x^7/7! + 701959131050*x^8/8! + 178186466260710*x^9/9! + 55669778154059882*x^10/10! + ...
%e A304641 exp(-x) * A(x) = 1 + x + 3*x^2/2! + 61*x^3/3! + 2811*x^4/4! + 214141*x^5/5! + 23949003*x^6/6! + 3665260621*x^7/7! + 732726498171*x^8/8! + 185070066199261*x^9/9! + 57591088296085803*x^10/10! + ...
%o A304641 (PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, (exp((m+1)*x +x*O(x^#A)) - Ser(A))^m ) )[#A] ); n!*A[n+1]}
%o A304641 for(n=0,20, print1(a(n),", "))
%Y A304641 Cf. A304640, A304642.
%K A304641 nonn
%O A304641 0,2
%A A304641 _Paul D. Hanna_, May 16 2018