cp's OEIS Frontend

A249409 E.g.f.: Sum_{n>=0} x^n * (exp(n*x) - 1)^n.

%I A249409 #11 Nov 05 2014 04:50:20
%S A249409 1,0,2,3,100,965,26166,652687,22791336,942744105,46394042410,
%T A249409 2686920755651,179344249073628,13675378594960381,1179389058183817950,
%U A249409 114010570608973463895,12267159489256485080656,1459502985288809274179153,190910980556047614953376594,27316077853047666256173322603
%N A249409 E.g.f.: Sum_{n>=0} x^n * (exp(n*x) - 1)^n.
%H A249409 Vaclav Kotesovec, <a href="/A249409/b249409.txt">Table of n, a(n) for n = 0..250</a>
%F A249409 E.g.f.: Sum_{n>=0} x^n * exp(n^2*x) / (1 + x*exp(n*x))^(n+1).
%F A249409 E.g.f.: Sum_{n>=0} x^n * exp(-n*x) / (x + exp(-n*x))^(n+1).
%e A249409 E.g.f.: A(x) = 1 + 2*x^2/2! + 3*x^3/3! + 100*x^4/4! + 965*x^5/5! +...
%e A249409 where the e.g.f. satisfies following series identity:
%e A249409 A(x) = 1 + x*(exp(x)-1) + x^2*(exp(2*x)-1)^2 + x^3*(exp(3*x)-1)^3 + x^4*(exp(4*x)-1)^4 + x^5*(exp(5*x)-1)^5 + x^6*(exp(6*x)-1)^6 +...
%e A249409 A(x) = 1/(1+x) + x*exp(x)/(1+x*exp(x))^2 + x^2*exp(4*x)/(1+x*exp(2*x))^3 + x^3*exp(9*x)/(1+x*exp(3*x))^4 + x^4*exp(16*x)/(1+x*exp(4*x))^5 + x^5*exp(25*x)/(1+x*exp(5*x))^6 + x^6*exp(36*x)/(1+x*exp(6*x))^7 +...
%e A249409 A(x) = 1/(x+1) + x*exp(-x)/(x+exp(-x))^2 + x^2*exp(-2*x)/(x+exp(-2*x))^3 + x^3*exp(-3*x)/(x+exp(-3*x))^4 + x^4*exp(-4*x)/(x+exp(-4*x))^5 + x^5*exp(-5*x)/(x+exp(-5*x))^6 + x^6*exp(-6*x)/(x+exp(-6*x))^7 +...
%o A249409 (PARI) {a(n)=local(A=1); A=sum(k=0, n, x^k * (exp(k*x +x*O(x^n)) - 1)^k); n!*polcoeff(A, n)}
%o A249409 for(n=0, 25, print1(a(n), ", "))
%o A249409 (PARI) {a(n)=local(A=1); A=sum(k=0, n, 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 A249409 for(n=0, 25, print1(a(n), ", "))
%o A249409 (PARI) {a(n)=local(A=1); A=sum(k=0, n, x^k * exp(-k*x +x*O(x^n)) / (x + exp(-k*x +x*O(x^n)))^(k+1) ); n!*polcoeff(A, n)}
%o A249409 for(n=0, 25, print1(a(n), ", "))
%Y A249409 Cf. A122399, A193421, A248615, A248653, A248654, A248655, A248471.
%K A249409 nonn
%O A249409 0,3
%A A249409 _Paul D. Hanna_, Oct 27 2014