A360482 E.g.f. satisfies A(x) = x * exp(x + 3 * A(x)).

0, 1, 8, 120, 2848, 92960, 3868224, 195810496, 11680512512, 802445898240, 62396469222400, 5417515922441216, 519519435065020416, 54535504354085687296, 6219954774471102242816, 765903524713482618101760, 101269330068289021683564544, 14310318526812295078276628480

Offset: 0

Author

Seiichi Manyama, Feb 09 2023

Keywords

• nonn

Links

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

Crossrefs

Cf. A216857, A360481, A360483, A360484.

Programs

PARI

my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(-3*x*exp(x))/3)))

PARI

a(n) = sum(k=1, n, 3^(k-1)*k^(n-1)*binomial(n, k));

Formula

E.g.f.: A(x) = -LambertW(-3*x * exp(x))/3.

a(n) = Sum_{k=1..n} 3^(k-1) * k^(n-1) * binomial(n,k).

a(n) ~ sqrt(1 + LambertW(exp(-1)/3)) * n^(n-1) /(3 * exp(n) * LambertW(exp(-1)/3)^n). - Vaclav Kotesovec, Feb 17 2023