OFFSET
0,2
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]
EXAMPLE
a(3) = 1557 = (1/e^3) * Sum_{j>=2} j^3 * 3^j / factorial(j-2).
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 *)
PROG
(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.
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Pedro Caceres, Oct 30 2020
STATUS
approved