A332188 a(n) = (1/e^n) * Sum_{j>=2} j^n * n^j / (j-2)!.

0, 3, 72, 1557, 36928, 986550, 29641608, 994006209, 36887753216, 1502798312547, 66730937637400, 3209318261685690, 166242143849148864, 9229638177763268395, 546842961612529341032, 34443269219453881669425

Offset: 0

Author

Pedro Caceres, Oct 30 2020

Keywords

• nonn

Examples

			a(3) = 1557 = (1/e^3) * Sum_{j>=2} j^3 * 3^j / factorial(j-2).
		

Crossrefs

Cf. A005494, A143495, A242817, A299824, A338282.

Programs

Mathematica

a[n_] := Sum[n^k*(StirlingS2[n + 2, k] - StirlingS2[n + 1, k]), {k, 2, n + 2}]; Array[a, 16, 0] (* Amiram Eldar, Oct 30 2020 *)

PARI

a(n) = sum(k=0, n+2, n^k*(stirling(n+2,k,2) - stirling(n+1,k,2))); \\ Michel Marcus, Oct 30 2020

SageMath

# Increase precision for larger n!
R = RealField(100)
t = 2
sol = [0]*18
for n in range(0, 18):
    suma = R(0)
    for j in range(t, 1000):
        suma += (j^n * n^j) / factorial(j - t)
    suma *= exp(-n)
    sol[n] = round(suma)
print(sol) # Thanks to Peter Luschny for his example in A338282.

Formula

a(n) = Sum_{k=0..n+2} n^k*(Stirling2(n+2,k) - Stirling2(n+1,k)). [Thanks to Andrew Howroyd for his example in A338282]