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%I #16 Oct 27 2014 22:38:41
%S 1,1,8,87,1836,49515,1888218,89221755,5280079752,373918542867,
%T 31364236227630,3055178634627603,341669063554284348,
%U 43366411236710160651,6190420280148645465330,985839231828100191119595,173934377807560257682643088,33790653299040390831598566051,7189209864606264796455405368022
%N E.g.f.: Sum_{n>=0} x^n * (3*exp(n*x) - 2)^n.
%F E.g.f.: Sum_{n>=0} (3*x)^n * exp(n^2*x) / (1 + 2*x*exp(n*x))^(n+1).
%e E.g.f.: A(x) = 1 + x + 8*x^2/2! + 87*x^3/3! + 1836*x^4/4! + 49515*x^5/5! +...
%e where the e.g.f. satisfies following series identity:
%e A(x) = 1 + x*(3*exp(x)-2) + x^2*(3*exp(2*x)-2)^2 + x^3*(3*exp(3*x)-2)^3 + x^4*(3*exp(4*x)-2)^4 + x^5*(3*exp(5*x)-2)^5 + x^6*(3*exp(6*x)-2)^6 +...
%e A(x) = 1/(1+2*x) + (3*x)*exp(x)/(1+2*x*exp(x))^2 + (3*x)^2*exp(4*x)/(1+2*x*exp(2*x))^3 + (3*x)^3*exp(9*x)/(1+2*x*exp(3*x))^4 + (3*x)^4*exp(16*x)/(1+2*x*exp(4*x))^5 + (3*x)^5*exp(25*x)/(1+2*x*exp(5*x))^6 + (3*x)^6*exp(36*x)/(1+2*x*exp(6*x))^7 +...
%o (PARI) {a(n)=local(A=1); A=sum(k=0, n, x^k * (3*exp(k*x +x*O(x^n)) - 2)^k); n!*polcoeff(A, n)}
%o for(n=0, 25, print1(a(n), ", "))
%o (PARI) {a(n)=local(A=1); A=sum(k=0, n, (3*x)^k * exp(k^2*x +x*O(x^n)) / (1 + 2*x*exp(k*x +x*O(x^n)))^(k+1) ); n!*polcoeff(A, n)}
%o for(n=0, 25, print1(a(n), ", "))
%Y Cf. A193421, A248615, A248653, A248654, A248655, A249409.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Oct 27 2014
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