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FORMULA
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E.g.f.: Sum_{n>=0} exp(n*(n+1)*x) / (1 + exp(n*x))^(n+1).
E.g.f.: Sum_{n>=0} 1 / (1 + exp(-n*x))^(n+1).
O.g.f.: Sum_{n>=0} (n+1)^n * n! * x^n / Product_{k=1..n} (1 - (n+1)*k*x).
a(n) = Sum_{k=0..n} (k+1)^n * k! * Stirling2(n,k).
a(n) ~ c * r^(2*n) * n^(2*n+1/2) * (1+exp(1/r))^n / exp(2*n), where r = 0.8737024332396683304965683047207192982139922672025395099... is the root of the equation (1+exp(-1/r)) * LambertW(-exp(-1/r)/r) = -1/r, and c = 6.4659886138749084757432110017013709735979825189027... . - Vaclav Kotesovec, Nov 07 2014
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EXAMPLE
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E.g.f.: E(x) = 1 + 2*x + 22*x^2/2! + 554*x^3/3! + 25366*x^4/4! + 1844042*x^5/5! +...
such that
E(x) = 1 + (exp(2*x)-1) + (exp(3*x)-1)^2 + (exp(4*x)-1)^3 + (exp(5*x)-1)^4 +...
The e.g.f. is also given by the infinite series:
E(x) = 1/2 + exp(2*x)/(1+exp(x))^2 + exp(6*x)/(1+exp(2*x))^3 + exp(12*x)/(1+exp(3*x))^4 + exp(20*x)/(1+exp(4*x))^5 + exp(30*x)/(1+exp(5*x))^6 +...
or, equivalently,
E(x) = 1/2 + 1/(1+exp(-x))^2 + 1/(1+exp(-2*x))^3 + 1/(1+exp(-3*x))^4 + 1/(1+exp(-4*x))^5 + 1/(1+exp(-5*x))^6 + 1/(1+exp(-6*x))^7 +...
ORDINARY GENERATING FUNCTION.
O.g.f.: A(x) = 1 + 2*x + 22*x^2 + 554*x^3 + 25366*x^4 + 1844042*x^5 +...
where
A(x) = 1 + 2*x/(1-2*x) + 3^2*2!*x^2/((1-3*1*x)*(1-3*2*x)) + 4^3*3!*x^3/((1-4*1*x)*(1-4*2*x)*(1-4*3*x)) + 5^4*4!*x^4/((1-5*1*x)*(1-5*2*x)*(1-5*3*x)*(1-5*4*x)) + 6^5*5!*x^5/((1-6*1*x)*(1-6*2*x)*(1-6*3*x)*(1-6*4*x)*(1-6*5*x)) +...
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PROG
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(PARI) /* By definition: */
{a(n)=n!*polcoeff(sum(k=0, n, (exp((k+1)*x +x*O(x^n)) - 1)^k), n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* From e.g.f. infinite series: */
\p100 \\ set precision
{A=Vec(serlaplace(sum(n=0, 500, 1.*exp(n*(n+1)*x +O(x^26))/(1 + exp(n*x +O(x^26)))^(n+1)) ))}
for(n=0, #A-1, print1(round(A[n+1]), ", "))
(PARI) /* From o.g.f.: */
{a(n)=polcoeff(sum(m=0, n, (m+1)^m*m!*x^m/prod(k=1, m, 1-(m+1)*k*x+x*O(x^n))), n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n)=sum(k=0, n, (k+1)^n * k! * Stirling2(n, k))}
for(n=0, 25, print1(a(n), ", "))
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