A195005 E.g.f.: Sum_{n>=0} 2^n*(exp(n*x) - 1)^n.

1, 2, 34, 1490, 122530, 16227602, 3155309794, 846406200530, 299510392317730, 135163342884412562, 75760096553546176354, 51633670624622762956370, 42049600429338786951232930, 40326932840083815683430101522, 44984263429111569097120217311714

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

Cf. A195263, A122399, A301581, A338040, A338044.

Sequence in context: A264669 A367510 A187546 * A215957 A171732 A291903

Adjacent sequences: A195002 A195003 A195004 * A195006 A195007 A195008

Keyword

nonn

Author

Paul D. Hanna, Sep 13 2011

Status

approved