A318615 -id:A318615 - cp's OEIS frontend

A290158 a(n) = n! * [x^n] exp(-n*x)/(1 + LambertW(-x)).

1, 0, 4, -9, 208, -1525, 33516, -463099, 11293248, -231839577, 6517863100, -175791146311, 5723314711632, -189288946716181, 7083626583237036, -275649085963046475, 11724766124450058496, -522717581675749841713, 24981438186138642481404

Offset: 0

Ilya Gutkovskiy, Oct 06 2017

sign

The n-th term of the n-th inverse binomial transform of A000312.

G. C. Greubel, Table of n, a(n) for n = 0..385
N. J. A. Sloane, Transforms

Cf. A000312, A069856, A290840, A318615, A356806.

Table[n! SeriesCoefficient[Exp[-n x]/(1 + LambertW[-x]), {x, 0, n}], {n, 0, 18}]

a(n) = (-1)^n*n!*sum(k=0, n\2, n^k*stirling(n-k, k, 2)/(n-k)!); \\ Seiichi Manyama, May 05 2023

a(n) ~ (-1)^n * n^n / (1 + LambertW(1)). - Vaclav Kotesovec, Oct 06 2017
From Seiichi Manyama, May 05 2023: (Start)
a(n) = (-1)^n * n! * [x^n] exp(n * x * (exp(x) - 1)).
a(n) = (-1)^n * n! * Sum_{k=0..floor(n/2)} n^k * Stirling2(n-k,k)/(n-k)!.
a(n) = [x^n] Sum_{k>=0} (k*x)^k / (1 + n*x)^(k+1).
a(n) = Sum_{k=0..n} (-n)^(n-k) * k^k * binomial(n,k). (End)

A362834 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = (-1)^n * n! * Sum_{j=0..floor(n/2)} k^j * Stirling1(n-j,j)/(n-j)!.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 4, 3, 0, 1, 0, 6, 6, 20, 0, 1, 0, 8, 9, 64, 90, 0, 1, 0, 10, 12, 132, 300, 594, 0, 1, 0, 12, 15, 224, 630, 2568, 4200, 0, 1, 0, 14, 18, 340, 1080, 6642, 20160, 34544, 0, 1, 0, 16, 21, 480, 1650, 13536, 55440, 193856, 316008, 0

Offset: 0

Seiichi Manyama, May 05 2023

			Square array begins:
  1,  1,   1,   1,    1,    1, ...
  0,  0,   0,   0,    0,    0, ...
  0,  2,   4,   6,    8,   10, ...
  0,  3,   6,   9,   12,   15, ...
  0, 20,  64, 132,  224,  340, ...
  0, 90, 300, 630, 1080, 1650, ...

Columns k=0..3 give: A000007, A066166, A053489, A053490.
Main diagonal gives A318615.
Cf. A361652.

T(n, k) = (-1)^n*n!*sum(j=0, n\2, k^j*stirling(n-j, j, 1)/(n-j)!);

E.g.f. of column k: 1/(1 - x)^(k*x).

A362835 Expansion of e.g.f. 1/(1 + LambertW(x * log(1-x))).

Original entry on oeis.org

1, 0, 2, 3, 56, 270, 4704, 43260, 814736, 11356632, 240848640, 4492204200, 108396245088, 2513538490320, 68878522931568, 1896787592514360, 58622475066067200, 1860520458522196800, 64297710768900261888, 2303738717704104464640

Offset: 0

Seiichi Manyama, May 05 2023

nonn

Cf. A318615, A362836.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(x*log(1-x)))))

a(n) = (-1)^n * n! * Sum_{k=0..floor(n/2)} k^k * Stirling1(n-k,k)/(n-k)!.

A318616 a(n) = n! * [x^n] (1 - x)^(n*x).

Original entry on oeis.org

1, 0, -4, -9, 160, 1350, -14904, -335160, 1796096, 125615448, 204300000, -64591072920, -735003528192, 41673388751280, 1113912529707264, -30043364514345000, -1703374149711298560, 17822402097051182400, 2856178489894627203072, 12394040043610922716800, -5255899207995216384000000

Offset: 0

Ilya Gutkovskiy, Aug 30 2018

sign

Cf. A007114, A008275, A318615.

Table[n! SeriesCoefficient[(1 - x)^(n x), {x, 0, n}], {n, 0, 20}]
Join[{1}, Table[n! Sum[(-1)^k n^(n - k) StirlingS1[k, n - k]/k!, {k, n}], {n, 20}]]

a(n) = n! * [x^n] exp(-n*x*Sum_{k>=1} x^k/k).

a(n) = n! * Sum_{k=0..n} (-1)^k*n^(n-k)*Stirling1(k,n-k)/k!.

cp's OEIS Frontend

A290158 a(n) = n! * [x^n] exp(-n*x)/(1 + LambertW(-x)).

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A362834 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = (-1)^n * n! * Sum_{j=0..floor(n/2)} k^j * Stirling1(n-j,j)/(n-j)!.

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A362835 Expansion of e.g.f. 1/(1 + LambertW(x * log(1-x))).

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A318616 a(n) = n! * [x^n] (1 - x)^(n*x).

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