A298373 -id:A298373 - cp's OEIS frontend

A334243 a(n) = exp(n) * Sum_{k>=0} (k + n)^n * (-n)^k / k!.

1, 0, -2, -3, 44, 245, -2346, -33278, 186808, 6888555, -6774910, -1986368439, -10227075420, 738830661296, 10363304656782, -327255834908715, -9380517430358288, 152180429032236325, 9132761207739810618, -46897839494116200918, -9833058047657527541220

Offset: 0

Ilya Gutkovskiy, Apr 19 2020

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Eric Weisstein's World of Mathematics, Bell Polynomial

Cf. A292866, A298373, A334240, A334241, A334242.

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

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

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

A307080 a(n) = exp(1) * Sum_{k>=0} (-1)^k(nk + 1)^n/k!.

Original entry on oeis.org

1, 0, -3, 19, 497, -1899, -489491, -15433676, 618450881, 120846851155, 7012261819901, -467816186167659, -175527285590430863, -20961845760818684812, 568194037748383908653, 898095630359015975379151, 220433074470274983356464897, 16144974747716546214909454181

Offset: 0

Ilya Gutkovskiy, Jun 24 2019

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Cf. A000587, A284860, A285065, A293037, A298373, A307066, A308645.

Table[Exp[1] Sum[(-1)^k (n k + 1)^n/k!, {k, 0, Infinity}], {n, 0, 17}]
Table[n! SeriesCoefficient[Exp[1 + x - Exp[n x]], {x, 0, n}], {n, 0, 17}]
Join[{1}, Table[Sum[Binomial[n, k] n^k BellB[k, -1], {k, 0, n}], {n, 1, 17}]]

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

a(n) = Sum_{k=0..n} binomial(n,k) * n^k * A000587(k).

cp's OEIS Frontend

A334243 a(n) = exp(n) * Sum_{k>=0} (k + n)^n * (-n)^k / k!.

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Formula

A307080 a(n) = exp(1) * Sum_{k>=0} (-1)^k(nk + 1)^n/k!.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

cp's OEIS Frontend

A334243 a(n) = exp(n) * Sum_{k>=0} (k + n)^n * (-n)^k / k!.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A307080 a(n) = exp(1) * Sum_{k>=0} (-1)^k*(n*k + 1)^n/k!.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A307080 a(n) = exp(1) * Sum_{k>=0} (-1)^k(nk + 1)^n/k!.