A355575 - OEIS

%I #123 Nov 24 2024 23:08:33

%S 1,0,0,6,24,120,1080,10080,120960,1874880,34473600,738460800,

%T 17982518400,489858969600,14834839219200,498452777222400,

%U 18583796335104000,768773914900992000,35220800475250790400,1779227869201400217600,98469904378626772992000

%N a(n) = n! * Sum_{k=0..floor(n/3)} k^(n - 3*k)/k!.

%H Seiichi Manyama, <a href="/A355575/b355575.txt">Table of n, a(n) for n = 0..323</a>

%F E.g.f.: Sum_{k>=0} x^(3*k) / (k! * (1 - k * x)).

%F a(n) ~ sqrt(Pi) * exp((n - 1/2)/LambertW(exp(3/4)*(2*n - 1)/8) - 2*n) * n^(2*n + 1/2) / (sqrt(1 + LambertW(exp(3/4)*(2*n - 1)/8)) * 2^(2*n + 1/2) * LambertW(exp(3/4)*(2*n - 1)/8)^n). - _Vaclav Kotesovec_, Oct 30 2022

%t Join[{1}, Table[n!*Sum[k^(n - 3*k)/k!, {k, 0, n/3}], {n, 1, 20}]] (* _Vaclav Kotesovec_, Oct 30 2022 *)

%o (PARI) a(n) = n!*sum(k=0, n\3, k^(n-3*k)/k!);

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^(3*k)/(k!*(1-k*x)))))

%Y Cf. A345747, A354436.

%Y Cf. A292889, A352945.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Sep 17 2022