OFFSET
0,2
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..217
FORMULA
a(n) = Sum_{k=0..n} 2^k*k^n*k!*Stirling2(n,k).
a(n) ~ c * (1 + 2*exp(1/r))^n * r^(2*n) * n!^2 / sqrt(n), where r = 0.925556278640887084941460444526398190071550948416... is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/2 and c = 0.3559088366632706316517829481255877447669425726507348... - Vaclav Kotesovec, Oct 04 2020
EXAMPLE
E.g.f.: A(x) = 1 + 2*x + 34*x^2/2! + 1490*x^3/3! + 122530*x^4/4! +...
where
A(x) = 1 + 2*(exp(x)-1) + 2^2*(exp(2*x)-1)^2 + 2^3*(exp(3*x)-1)^3 +...
MATHEMATICA
Flatten[{1, Table[Sum[2^k * k^n * k! * StirlingS2[n, k], {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Oct 04 2020 *)
PROG
(PARI) {a(n)=local(X=x+x*O(x^n)); n!*polcoeff(sum(m=0, n, 2^m*(exp(m*X)-1)^m), n)}
(PARI) {Stirling2(n, k)=if(k<0|k>n, 0, sum(i=0, k, (-1)^i*binomial(k, i)/k!*(k-i)^n))}
{a(n)=sum(k=0, n, 2^k*k^n*k!*Stirling2(n, k))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 13 2011
STATUS
approved