A337012 a(n) = exp(-1/2) * Sum_{k>=0} (2*k + n)^n / (2^k * k!).

1, 2, 11, 92, 1025, 14232, 236403, 4568720, 100670529, 2490511776, 68341981051, 2059882505408, 67645498798721, 2403948686290816, 91914992104815459, 3762299973887526144, 164148252324092964993, 7604537914425558921728, 372812121514187124192875

Offset: 0

Author

Ilya Gutkovskiy, Aug 11 2020

Keywords

• nonn

Crossrefs

Cf. A004211, A007405, A134980, A337010, A337011.

Programs

Mathematica

Table[n! SeriesCoefficient[Exp[n x + (Exp[2 x] - 1)/2], {x, 0, n}], {n, 0, 18}]
Unprotect[Power]; 0^0 = 1; Table[Sum[Binomial[n, k] n^(n - k) 2^k BellB[k, 1/2], {k, 0, n}], {n, 0, 18}]

Formula

a(n) = n! * [x^n] exp(n*x + (exp(2*x) - 1) / 2).

a(n) = Sum_{k=0..n} binomial(n,k) * n^(n-k) * A004211(k).