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%I #3 Oct 18 2014 12:41:17
%S 1,4,34,483,10084,286885,10556406,483876799,26866889512,1768601369961,
%T 135698985275050,11968589697570451,1199598182911257372,
%U 135313142875442335453,17035239326998414091038,2376497634554143028502855,365070055205852728328220496,61412309543674687202717299921
%N E.g.f.: Sum_{n>=0} x^n * (3 + exp(n*x))^n.
%F E.g.f.: Sum_{n>=0} x^n * exp(n^2*x)/(1 - 3*x*exp(n*x))^(n+1).
%e G.f.: A(x) = 1 + 4*x + 34*x^2/2! + 483*x^3/3! + 10084*x^4/4! + 286885*x^5/5! +...
%e where the g.f. satisfies following series identity:
%e A(x) = 1 + x*(3+exp(x)) + x^2*(3+exp(2*x))^2 + x^3*(3+exp(3*x))^3 + x^4*(3+exp(4*x))^4 + x^5*(3+exp(5*x))^5 + x^6*(3+exp(6*x))^6 +...
%e A(x) = 1/(1-3*x) + x*exp(x)/(1-3*x*exp(x))^2 + x^2*exp(4*x)/(1-3*x*exp(2*x))^3 + x^3*exp(9*x)/(1-3*x*exp(3*x))^4 + x^4*exp(16*x)/(1-3*x*exp(4*x))^5 + x^5*exp(25*x)/(1-3*x*exp(5*x))^6 + x^6*exp(36*x)/(1-3*x*exp(6*x))^7 +...
%o (PARI) {a(n,t=3)=local(A=1); A=sum(k=0, n, x^k * (t + exp(k*x +x*O(x^n)))^k); n!*polcoeff(A, n)}
%o for(n=0,25,print1(a(n,3),", "))
%o (PARI) {a(n,t=3)=local(A=1); A=sum(k=0, n, x^k * exp(k^2*x +x*O(x^n))/(1 - t*x*exp(k*x +x*O(x^n)))^(k+1) ); n!*polcoeff(A, n)}
%o for(n=0,25,print1(a(n,3),", "))
%Y Cf. A248615, A248653, A193421.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Oct 18 2014
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