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%I #9 Feb 24 2014 16:09:47
%S 1,1,2,9,64,630,8030,126247,2371612,52026293,1309661828,37318672196,
%T 1190672836246,42159850045181,1644546319080848,70233352188006641,
%U 3266637689293293616,164720219739258021686,8969422973088951968070,525585300443124229026511,33039986976855724686082476
%N E.g.f. satisfies: A(x) = Sum_{n>=0} Integral( A(x)^n dx )^n/n!, where the constant of integration is zero.
%C Compare to the identity:
%C if G(x) = Sum_{n>=0} Integral( G(x)^t dx )^n/n!, then G(x)^t = 1/(1 - t*x).
%H Vaclav Kotesovec, <a href="/A232552/b232552.txt">Table of n, a(n) for n = 0..100</a>
%F E.g.f. satisfies: A'(x) = Sum_{n>=1} A(x)^n * ( Integral A(x)^n dx )^(n-1)/(n-1)!.
%e G.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 64*x^4/4! + 630*x^5/5! + 8030*x^6/6! +...
%e Let B(n,x) = Integral( A(x)^n dx ) with B(n,0)=0, then
%e A(x) = 1 + B(1,x) + B(2,x)^2/2! + B(3,x)^3/3! + B(4,x)^4/4! + B(5,x)^5/5! +...
%e A'(x) = A(x) + A(x)^2*B(2,x) + A(x)^3*B(3,x)^2/2! + A(x)^4*B(4,x)^3/3! +...
%e where
%e B(1,x) = x + x^2/2! + 2*x^3/3! + 9*x^4/4! + 64*x^5/5! + 630*x^6/6! +...
%e B(2,x) = x + 2*x^2/2! + 6*x^3/3! + 30*x^4/4! + 224*x^5/5! + 2260*x^6/6! +...
%e B(3,x) = x + 3*x^2/2! + 12*x^3/3! + 69*x^4/4! + 552*x^5/5! + 5790*x^6/6! +...
%e B(4,x) = x + 4*x^2/2! + 20*x^3/3! + 132*x^4/4! + 1144*x^5/5! + 12600*x^6/6! +...
%e B(5,x) = x + 5*x^2/2! + 30*x^3/3! + 225*x^4/4! + 2120*x^5/5! + 24670*x^6/6! +...
%e B(6,x) = x + 6*x^2/2! + 42*x^3/3! + 354*x^4/4! + 3624*x^5/5! + 44700*x^6/6! +...
%e ...
%e B(2,x)^2/2! = x^2/2! + 6*x^3/3! + 36*x^4/4! + 270*x^5/5! + 2604*x^6/6! +...
%e B(3,x)^3/3! = x^3/3! + 18*x^4/4! + 255*x^5/5! + 3600*x^6/6! + 54747*x^7/7! +...
%e B(4,x)^4/4! = x^4/4! + 40*x^5/5! + 1120*x^6/6! + 28140*x^7/7! + 693504*x^8/8! +...
%e B(5,x)^5/5! = x^5/5! + 75*x^6/6! + 3675*x^7/7! + 152250*x^8/8! + 5866245*x^9/9! +...
%e B(6,x)^6/6! = x^6/6! + 126*x^7/7! + 9912*x^8/8! + 634284*x^9/9! + 36483048*x^10/10! +...
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,intformal(A^m + x*O(x^n))^m/m!));n!*polcoeff(A,n)}
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A234855.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 26 2013
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