A354436 a(n) = n! * Sum_{k=0..n} k^(n-k)/k!.

1, 1, 3, 13, 85, 801, 10231, 168253, 3437673, 85162465, 2511412651, 86805640461, 3469622549053, 158523442439233, 8198514736542495, 476003264246418301, 30804251925861439441, 2207978115389469465153, 174304316334466458575443

Offset: 0

Author

Seiichi Manyama, May 28 2022

Keywords

• nonn

Crossrefs

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

Programs

Mathematica

Join[{1}, Table[n!*Sum[k^(n-k)/k!, {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, May 28 2022 *)

PARI

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

PARI

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

Python

from math import factorial
def A354436(n): return sum(factorial(n)*k**(n-k)//factorial(k) for k in range(n+1)) # Chai Wah Wu, May 28 2022

Formula

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

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

a(n) = Sum_{k=0..n} (n-k)^k*k!*binomial(n,k). - Ridouane Oudra, Jun 17 2025