A338282 a(n) = (1/e^n) * Sum_{j>=3} j^n * n^j / (j-3)!.

0, 4, 216, 7371, 239424, 8127875, 296315496, 11685617608, 498593804800, 22959117809685, 1137033860419000, 60338078785131785, 3418430599382500800, 206053517402599981504, 13172124530670958537160, 890361160360138336174875, 63463906792476058870550528, 4758276450884470061869230823

Offset

0,2

Links

Table of n, a(n) for n=0..17.

Formula

a(n) = Sum_{k=0..n+3} n^k*(Stirling2(n+3,k) - 3*Stirling2(n+2,k) + 2*Stirling2(n+1,k)). - Andrew Howroyd, Oct 20 2020

a(n) = Sum_{k=0..n} n^(k+3)*A143495(n+3, k+3). - Peter Luschny, Oct 21 2020

Example

a(3) = 7371 = (1/e^3) * Sum_{j>=3} j^3 * 3^j / factorial(j-3).

Maple

seq(add(n^(k+3)*A143495(n+3, k+3), k = 0..n), n = 0..17); # Peter Luschny, Oct 21 2020

Mathematica

a[n_] := Exp[-n] * Sum[j^n * n^j/(j - 3)!, {j, 3, Infinity}]; Array[a, 17, 0] (* Amiram Eldar, Oct 20 2020 *)

Prog

(SageMath) # Increase precision for larger n!
R = RealField(100)
t = 3
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) # Peter Luschny, Oct 20 2020
(PARI) a(n)={sum(k=0, n+3, n^k*(stirling(n+3, k, 2) - 3*stirling(n+2, k, 2) + 2*stirling(n+1, k, 2)))} \\ Andrew Howroyd, Oct 20 2020

Crossrefs

Cf. A005494, A143495, A242817, A299824.

Sequence in context: A038790 A042325 A091287 * A281997 A276487 A269283

Adjacent sequences: A338279 A338280 A338281 * A338283 A338284 A338285

Keyword

nonn

Author

Pedro Caceres, Oct 20 2020

Status

approved