A337057 a(n) = exp(-n) * Sum_{k>=0} (k - n)^n * n^k / k!.

1, 0, 2, 3, 52, 255, 4146, 38766, 688584, 9685017, 195875110, 3655101703, 84872077500, 1955205893680, 51896551499898, 1412668946049315, 42475968202854160, 1328074354724554471, 44778480417250291566, 1577210136570598631318

Offset: 0

Author

Ilya Gutkovskiy, Aug 13 2020

Keywords

• nonn

Crossrefs

Cf. A000296, A194689, A242817, A334242, A335867.

Programs

Mathematica

Table[n! SeriesCoefficient[Exp[n (Exp[x] - 1 - x)], {x, 0, n}], {n, 0, 19}]
Unprotect[Power]; 0^0 = 1; Table[Sum[Binomial[n, k] (-n)^(n - k) BellB[k, n], {k, 0, n}], {n, 0, 19}]

Formula

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

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