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%I #3 Nov 15 2012 21:25:16
%S 1,1,32,3119,625710,214333471,112105268136,83149960883200,
%T 83014425998481126,107334569041127441462,174471878478682785998864,
%U 348242875992753988109552778,837327855535084109106340786272,2387108242583316451939303856237037
%N O.g.f. satisfies: A(x) = Sum_{n>=0} A(x)^n * (n^3*x)^n/n! * exp(-n^3*x*A(x)).
%C Compare to the o.g.f. of A217913:
%C Sum_{n>=0} (n^3)^n*exp(-n^3*x)*x^n/n! = Sum_{n>=0} Stirling2(3*n,n)*x^n.
%F O.g.f. satisfies: A(x) = Sum_{n>=0} Stirling2(3*n,n) * x^n * A(x)^n.
%e O.g.f.: A(x) = 1 + x + 32*x^2 + 3119*x^3 + 625710*x^4 + 214333471*x^5 +...
%e where
%e A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^6*x^2*A(x)^2/2!*exp(-8*x*A(x)) + 3^9*x^3*A(x)^3/3!*exp(-27*x*A(x)) + 4^12*x^4*A(x)^4/4!*exp(-64*x*A(x)) + 5^15*x^5*A(x)^5/5!*exp(-125*x*A(x)) +...
%e simplifies to a power series in x with integer coefficients.
%e G.f. A(x) satisfies A(x) = G(x*A(x)) where G(x) = A(x/G(x)) begins:
%e G(x) = 1 + x + 31*x^2 + 3025*x^3 + 611501*x^4 + 210766920*x^5 + 110687251039*x^6 +...+ Stirling2(3*n,n)*x^n +...
%o (PARI) {a(n)=local(A=1);for(i=1,n,A=sum(m=0, n, (m^3*x*A)^m/m!*exp(-m^3*x*A+x*O(x^n))));polcoeff(A, n)}
%o for(n=0,21,print1(a(n),", "))
%Y Cf. A217913, A219184, A217900, A218681, A218672.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 15 2012
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