A354436 - OEIS

%I #20 May 29 2022 01:58:24

%S 1,1,3,13,85,801,10231,168253,3437673,85162465,2511412651,86805640461,

%T 3469622549053,158523442439233,8198514736542495,476003264246418301,

%U 30804251925861439441,2207978115389469465153,174304316334466458575443

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

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

%F 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

%t Join[{1}, Table[n!*Sum[k^(n-k)/k!, {k, 0, n}], {n, 1, 20}]] (* _Vaclav Kotesovec_, May 28 2022 *)

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

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

%o (Python)

%o from math import factorial

%o 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 Cf. A006153, A026898, A010844, A277452, A277506, A354437.

%K nonn

%O 0,3

%A _Seiichi Manyama_, May 28 2022