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 A356629 #21 Nov 01 2022 12:10:35
%S A356629 1,1,1,1,25,121,361,5881,82321,547345,6053041,167991121,2179469161,
%T A356629 22892967241,788375451865,18046198202761,245523704069281,
%U A356629 7548055281543841,270833271588545761,5369819950838359585,141456920470310708281,6760255576117937586841
%N A356629 a(n) = n! * Sum_{k=0..floor(n/3)} (n - 3*k)^k/(n - 3*k)!.
%H A356629 Seiichi Manyama, <a href="/A356629/b356629.txt">Table of n, a(n) for n = 0..393</a>
%F A356629 E.g.f.: Sum_{k>=0} x^k / (k! * (1 - k*x^3)).
%F A356629 a(n) ~ sqrt(Pi/3) * exp((2*n - 3)/(6*LambertW(exp(1/4)*(2*n - 3)/8)) - 4*n/3) * n^(4*n/3 + 1/2) / (sqrt(1 + LambertW(exp(1/4)*(2*n - 3)/8)) * 2^(2*n/3 + 1/2) * LambertW(exp(1/4)*(2*n - 3)/8)^(n/3)). - _Vaclav Kotesovec_, Nov 01 2022
%t A356629 a[n_] := n! * Sum[(n - 3*k)^k/(n - 3*k)!, {k, 0, Floor[n/3]}]; a[0] = 1; Array[a, 22, 0] (* _Amiram Eldar_, Aug 19 2022 *)
%o A356629 (PARI) a(n) = n!*sum(k=0, n\3, (n-3*k)^k/(n-3*k)!);
%o A356629 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(k!*(1-k*x^3)))))
%Y A356629 Cf. A354436, A356628, A356630.
%Y A356629 Cf. A354553, A356633, A358065.
%K A356629 nonn
%O A356629 0,5
%A A356629 _Seiichi Manyama_, Aug 18 2022