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E.g.f. A(x) satisfies: 1 - x = Sum_{n>=0} (Integral -A(x)^n dx)^n / n!.
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%I #11 Feb 17 2018 05:19:52

%S 1,1,5,43,514,7778,141427,2990741,71982197,1943318293,58274530808,

%T 1928170445532,70050754418969,2781721368620685,120104488614903489,

%U 5601368827311472211,280010977440133986954,14884513825530929143826,835402492859596917623243,49259935601350475622014329,3045801421611820426020858069

%N E.g.f. A(x) satisfies: 1 - x = Sum_{n>=0} (Integral -A(x)^n dx)^n / n!.

%H Paul D. Hanna, <a href="/A299425/b299425.txt">Table of n, a(n) for n = 0..100</a>

%F E.g.f. A(x) satisfies:

%F (1) 1 - x = Sum_{n>=0} (Integral -A(x)^n dx)^n / n!.

%F (2) 1 = Sum_{n>=0} A(x)^(n+1) * (Integral -A(x)^(n+1) dx)^n / n!.

%e E.g.f.: A(x) = 1 + x + 5*x^2/2! + 43*x^3/3! + 514*x^4/4! + 7778*x^5/5! + 141427*x^6/6! + 2990741*x^7/7! + 71982197*x^8/8! + 1943318293*x^9/9! + 58274530808*x^10/10! + ...

%e such that

%e 1 - x = 1 - (Integral A(x) dx) + (Integral A(x)^2 dx)^2/2! - (Integral A(x)^3 dx)^3/3! + (Integral A(x)^4 dx)^4/4! - (Integral A(x)^5 dx)^5/5! + ...

%e also

%e 1 = A(x) - A(x)^2*(Integral A(x)^2 dx) + A(x)^3*(Integral A(x)^3 dx)^2/2! - A(x)^4*(Integral A(x)^4 dx)^3/3! + A(x)^5*(Integral A(x)^5 dx)^4/4! + ...

%e Related series.

%e A'(x) = F(x)/G(x) where

%e F(x) = Sum_{n>=0} A(x)^(2*n+4) * (Integral -A(x)^(n+2) dx)^n / n! and

%e G(x) = Sum_{n>=0} (n+1) * A(x)^n * (Integral -A(x)^(n+1) dx)^n / n!.

%o (PARI) {a(n) = my(A=1); for(i=1, n, A = 1 - sum(m=1, n, A^(m+1) * intformal( -A^(m+1) +x*O(x^n) )^m/m!)); n!*polcoeff(A,n)}

%o for(n=0, 20, print1(a(n), ", "))

%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); A[#A] = -Vec( sum(m=0,#A+1, intformal( -Ser(A)^(m+1) )^m/m! * Ser(A)^(m+1)) )[#A] ); n!*A[n+1]}

%o for(n=0,20,print1(a(n),", "))

%K sign

%O 0,3

%A _Paul D. Hanna_, Feb 15 2018