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%I #6 Jan 28 2016 11:57:55
%S 1,1,2,5,16,65,326,1947,13410,104181,900214,8566655,89055224,
%T 1004141647,12204369138,159036267519,2211764983734,32696763676521,
%U 511987792322430,8465194670035767,147370831072230860,2694506417687396995,51622643862824956898,1034153511794063402519,21621325640846679627146
%N E.g.f. A(x) satisfies: A(x) = exp( Integral B(x) dx ) such that B(x) = exp(1+x - exp(x)) * exp( Integral A(x) dx ), where the constant of integration is zero.
%C Compare to: F(x) = exp( Integral G(x) dx ) such that G(x) = exp(1-exp(x)) * exp( Integral F(x) dx ) holds when F(x) = exp(x).
%e E.g.f.: A(x) = 1 + x + 2*x^2/2! + 5*x^3/3! + 16*x^4/4! + 65*x^5/5! + 326*x^6/6! + 1947*x^7/7! + 13410*x^8/8! + 104181*x^9/9! + 900214*x^10/10! + 8566655*x^11/11! +...
%e such that log(A(x)) = Integral B(x) dx
%e where
%e B(x) = 1 + x + x^2/2! + 2*x^3/3! + 9*x^4/4! + 46*x^5/5! + 245*x^6/6! + 1474*x^7/7! + 10315*x^8/8! + 82174*x^9/9! + 726591*x^10/10! + 7038632*x^11/11! + 74216949*x^12/12! +...+ A268171(n)*x^n/n! +...
%e and A(x) and B(x) satisfy:
%e (1) A(x) = B'(x)/B(x) + exp(x) - 1,
%e (2) B(x) = A'(x)/A(x),
%e (3) log(A(x)) = Integral B(x) dx,
%e (4) log(B(x)) = Integral A(x) dx + 1+x - exp(x).
%e RELATED SERIES.
%e log(A(x)) = x + x^2/2! + x^3/3! + 2*x^4/4! + 9*x^5/5! + 46*x^6/6! + 245*x^7/7! + 1474*x^8/8! + 10315*x^9/9! + 82174*x^10/10! + 726591*x^11/11! + 7038632*x^12/12! +...
%e Let J(x) equal the series reversion of log(A(x)); then
%e J(x) = x - x^2/2! + 2*x^3/3! - 7*x^4/4! + 31*x^5/5! - 172*x^6/6! + 1155*x^7/7! - 9027*x^8/8! + 80676*x^9/9! - 811727*x^10/10! + 9075333*x^11/11! - 111633356*x^12/12! +...
%e where A(J(x)) = exp(x).
%o (PARI) {a(n) = my(A=1+x, B=1+x); for(i=0, n, A = exp( intformal( B + x*O(x^n) ) ); B = exp(1+x - exp(x +x*O(x^n)) + intformal( A ) ) ); n!*polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A266328, A266329, A266490, A268171.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 27 2016
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