A335867 -id:A335867 - cp's OEIS frontend

A335868 a(n) = exp(n) * Sum_{k>=0} (-n)^k * (k - 1)^n / k!.

1, -2, 7, -31, 149, -631, 475, 43210, -844727, 10960505, -86569889, -584746911, 46302579229, -1304510879686, 25366896568707, -277053418780891, -4271166460501743, 384590020131637825, -14617527176248527545, 380117694164438489422, -5265650620303861935579

Offset: 0

Ilya Gutkovskiy, Jun 27 2020

sign

Seiichi Manyama, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Bell Polynomial

Cf. A109747, A292866, A334241, A335867.

Table[n! SeriesCoefficient[Exp[n (1 - Exp[x]) - x], {x, 0, n}], {n, 0, 20}]
Table[Sum[(-1)^(n - k) Binomial[n, k] BellB[k, -n], {k, 0, n}], {n, 0, 20}]

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

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * BellPolynomial_k(-n).

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

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Aug 13 2020

nonn

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

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}]

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).

cp's OEIS Frontend

A335868 a(n) = exp(n) * Sum_{k>=0} (-n)^k * (k - 1)^n / k!.

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