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

%I A338282 #37 Oct 21 2020 11:24:28
%S A338282 0,4,216,7371,239424,8127875,296315496,11685617608,498593804800,
%T A338282 22959117809685,1137033860419000,60338078785131785,
%U A338282 3418430599382500800,206053517402599981504,13172124530670958537160,890361160360138336174875,63463906792476058870550528,4758276450884470061869230823
%N A338282 a(n) = (1/e^n) * Sum_{j>=3} j^n * n^j / (j-3)!.
%F A338282 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
%F A338282 a(n) = Sum_{k=0..n} n^(k+3)*A143495(n+3, k+3). - _Peter Luschny_, Oct 21 2020
%e A338282 a(3) = 7371 = (1/e^3) * Sum_{j>=3} j^3 * 3^j / factorial(j-3).
%p A338282 seq(add(n^(k+3)*A143495(n+3, k+3), k = 0..n), n = 0..17); # _Peter Luschny_, Oct 21 2020
%t A338282 a[n_] := Exp[-n] * Sum[j^n * n^j/(j - 3)!, {j, 3, Infinity}]; Array[a, 17, 0] (* _Amiram Eldar_, Oct 20 2020 *)
%o A338282 (SageMath) # Increase precision for larger n!
%o A338282 R = RealField(100)
%o A338282 t = 3
%o A338282 sol = [0]*18
%o A338282 for n in range(0, 18):
%o A338282     suma = R(0)
%o A338282     for j in range(t, 1000):
%o A338282         suma += (j^n * n^j) / factorial(j - t)
%o A338282     suma *= exp(-n)
%o A338282     sol[n] = round(suma)
%o A338282 print(sol) # _Peter Luschny_, Oct 20 2020
%o A338282 (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
%Y A338282 Cf. A005494, A143495, A242817, A299824.
%K A338282 nonn
%O A338282 0,2
%A A338282 _Pedro Caceres_, Oct 20 2020