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A248655 E.g.f.: Sum_{n>=0} x^n * (2*exp(n*x) - 1)^n.

%I A248655 #7 Nov 05 2014 04:51:14
%S A248655 1,1,6,60,1040,24650,796332,32556650,1650306352,100688420034,
%T A248655 7262453789300,609263939139482,58684327404118440,6419194423681262210,
%U A248655 790005709370352467020,108514436287885023435930,16520198768574933272377952,2770418328864442110268310402,508995643667831746279224633828
%N A248655 E.g.f.: Sum_{n>=0} x^n * (2*exp(n*x) - 1)^n.
%H A248655 Vaclav Kotesovec, <a href="/A248655/b248655.txt">Table of n, a(n) for n = 0..160</a>
%F A248655 E.g.f.: Sum_{n>=0} (2*x)^n * exp(n^2*x) / (1 + x*exp(n*x))^(n+1).
%e A248655 E.g.f.: A(x) = 1 + x + 6*x^2/2! + 60*x^3/3! + 1040*x^4/4! + 24650*x^5/5! +...
%e A248655 where the e.g.f. satisfies following series identity:
%e A248655 A(x) = 1 + x*(2*exp(x)-1) + x^2*(2*exp(2*x)-1)^2 + x^3*(2*exp(3*x)-1)^3 + x^4*(2*exp(4*x)-1)^4 + x^5*(2*exp(5*x)-1)^5 + x^6*(2*exp(6*x)-1)^6 +...
%e A248655 A(x) = 1/(1+x) + (2*x)*exp(x)/(1+x*exp(x))^2 + (2*x)^2*exp(4*x)/(1+x*exp(2*x))^3 + (2*x)^3*exp(9*x)/(1+x*exp(3*x))^4 + (2*x)^4*exp(16*x)/(1+x*exp(4*x))^5 + (2*x)^5*exp(25*x)/(1+x*exp(5*x))^6 + (2*x)^6*exp(36*x)/(1+x*exp(6*x))^7 +...
%o A248655 (PARI) {a(n)=local(A=1); A=sum(k=0, n, x^k * (2*exp(k*x +x*O(x^n)) - 1)^k); n!*polcoeff(A, n)}
%o A248655 for(n=0,25,print1(a(n),", "))
%o A248655 (PARI) {a(n)=local(A=1); A=sum(k=0, n, (2*x)^k * exp(k^2*x +x*O(x^n)) / (1 + x*exp(k*x +x*O(x^n)))^(k+1) ); n!*polcoeff(A, n)}
%o A248655 for(n=0,25,print1(a(n),", "))
%Y A248655 Cf. A193421, A248615, A248653, A248654.
%K A248655 nonn
%O A248655 0,3
%A A248655 _Paul D. Hanna_, Oct 26 2014