A360548 E.g.f. satisfies A(x) = x * exp( 2*(x + A(x)) ).

0, 1, 8, 96, 1792, 46080, 1511424, 60325888, 2837970944, 153778913280, 9432255692800, 646039266656256, 48874810528235520, 4047655951598092288, 364221261622538141696, 35384754572803304325120, 3691411033400626898796544, 411569264258973944034361344

Offset: 0

Author

Seiichi Manyama, Feb 11 2023

Keywords

• nonn

Links

• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Cf. A100526, A216857, A360545.

Cf. A038049, A201595, A214225, A360547.

Programs

Maple

A360548 := proc(n)
    add((2*k)^(n-1)*binomial(n,k),k=1..n) ;
end proc:
seq(A360548(n),n=0..60) ; # R. J. Mathar, Mar 12 2023

PARI

my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(-2*x*exp(2*x))/2)))

PARI

a(n) = sum(k=1, n, (2*k)^(n-1)*binomial(n, k));

Formula

E.g.f.: A(x) = (-1/2) * LambertW(-2*x * exp(2*x)).

a(n) = Sum_{k=1..n} (2*k)^(n-1) * binomial(n,k) = 4^(n-1) * A100526(n).

a(n) ~ sqrt(1 + LambertW(exp(-1))) * 2^(n-1) * n^(n-1) / (LambertW(exp(-1))^n * exp(n)). - Vaclav Kotesovec, Feb 17 2023