A248615 E.g.f.: Sum_{n>=0} x^n * (1 + exp(n*x))^n.

1, 2, 10, 99, 1540, 33765, 978486, 35904463, 1617090728, 87295834665, 5543664590890, 407913496939971, 34345772643864540, 3274503583293002749, 350340544319075248862, 41739496421354682909015, 5500282856601438305701456, 796938625536899521586167377, 126293050155194034442298968914

Offset: 0

Author

Paul D. Hanna, Oct 09 2014

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + 2*x + 10*x^2/2! + 99*x^3/3! + 1540*x^4/4! + 33765*x^5/5! +...
where the e.g.f. satisfies following series identity:
A(x) = 1 + x*(1+exp(x)) + x^2*(1+exp(2*x))^2 + x^3*(1+exp(3*x))^3 + x^4*(1+exp(4*x))^4 + x^5*(1+exp(5*x))^5 + x^6*(1+exp(6*x))^6 +...
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 +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..260

Crossrefs

Cf. A193421.

Programs

PARI

{a(n, t=1)=local(A=1+x); A=sum(k=0, n, x^k * (t + exp(k*x +x*O(x^n)) +x*O(x^n))^k); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n,1), ", "))

PARI

{a(n, t=1)=local(A=1+x); A=sum(k=0, n, exp(k^2*x +x*O(x^n))*x^k/(1 - t*exp(k*x +x*O(x^n))*x )^(k+1) ); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n,1), ", "))

Formula

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