This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A355575 #123 Nov 24 2024 23:08:33
%S A355575 1,0,0,6,24,120,1080,10080,120960,1874880,34473600,738460800,
%T A355575 17982518400,489858969600,14834839219200,498452777222400,
%U A355575 18583796335104000,768773914900992000,35220800475250790400,1779227869201400217600,98469904378626772992000
%N A355575 a(n) = n! * Sum_{k=0..floor(n/3)} k^(n - 3*k)/k!.
%H A355575 Seiichi Manyama, <a href="/A355575/b355575.txt">Table of n, a(n) for n = 0..323</a>
%F A355575 E.g.f.: Sum_{k>=0} x^(3*k) / (k! * (1 - k * x)).
%F A355575 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 A355575 Join[{1}, Table[n!*Sum[k^(n - 3*k)/k!, {k, 0, n/3}], {n, 1, 20}]] (* _Vaclav Kotesovec_, Oct 30 2022 *)
%o A355575 (PARI) a(n) = n!*sum(k=0, n\3, k^(n-3*k)/k!);
%o A355575 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^(3*k)/(k!*(1-k*x)))))
%Y A355575 Cf. A345747, A354436.
%Y A355575 Cf. A292889, A352945.
%K A355575 nonn
%O A355575 0,4
%A A355575 _Seiichi Manyama_, Sep 17 2022