A360545 - OEIS

%I #19 Feb 16 2025 08:34:04

%S 0,1,6,54,756,14580,358668,10736712,378823392,15395255280,

%T 708217959600,36380741745744,2064234271203360,128214974795177088,

%U 8652900673357097472,630483717450225530880,49330027417316557012992,4124992361928178722764544

%N E.g.f. satisfies A(x) = x * exp( 3*(x + A(x))/2 ).

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.

%F E.g.f.: A(x) = (-2/3) * LambertW(-3*x/2 * exp(3*x/2)).

%F a(n) = Sum_{k=1..n} (3*k/2)^(n-1) * binomial(n,k) = 3^(n-1) * A100526(n).

%F a(n) ~ sqrt(1 + LambertW(exp(-1))) * 3^(n-1) * n^(n-1) / (2^(n-1) * exp(n) * LambertW(exp(-1))^n). - _Vaclav Kotesovec_, Feb 17 2023

%o (PARI) my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-2*lambertw(-3*x/2*exp(3*x/2))/3)))

%o (PARI) a(n) = sum(k=1, n, (3*k/2)^(n-1)*binomial(n, k));

%Y Cf. A100526, A216857, A360548.

%Y Cf. A360482, A360544.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Feb 11 2023