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 A354436 #23 Jun 17 2025 03:13:54
%S A354436 1,1,3,13,85,801,10231,168253,3437673,85162465,2511412651,86805640461,
%T A354436 3469622549053,158523442439233,8198514736542495,476003264246418301,
%U A354436 30804251925861439441,2207978115389469465153,174304316334466458575443
%N A354436 a(n) = n! * Sum_{k=0..n} k^(n-k)/k!.
%F A354436 E.g.f.: Sum_{k>=0} x^k / (k! * (1 - k*x)).
%F A354436 a(n) ~ sqrt(Pi) * exp((2*n-1)/(2*LambertW(exp(1/2)*(2*n-1)/4)) - 2*n) * n^(2*n + 1/2) / (sqrt(1 + LambertW(exp(1/2)*(2*n-1)/4)) * 2^n * LambertW(exp(1/2)*(2*n-1)/4)^n). - _Vaclav Kotesovec_, May 28 2022
%F A354436 a(n) = Sum_{k=0..n} (n-k)^k*k!*binomial(n,k). - _Ridouane Oudra_, Jun 17 2025
%t A354436 Join[{1}, Table[n!*Sum[k^(n-k)/k!, {k, 0, n}], {n, 1, 20}]] (* _Vaclav Kotesovec_, May 28 2022 *)
%o A354436 (PARI) a(n) = n!*sum(k=0, n, k^(n-k)/k!);
%o A354436 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(k!*(1-k*x)))))
%o A354436 (Python)
%o A354436 from math import factorial
%o A354436 def A354436(n): return sum(factorial(n)*k**(n-k)//factorial(k) for k in range(n+1)) # _Chai Wah Wu_, May 28 2022
%Y A354436 Cf. A006153, A026898, A010844, A277452, A277506, A354437.
%K A354436 nonn
%O A354436 0,3
%A A354436 _Seiichi Manyama_, May 28 2022