A290840 -id:A290840 - 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)

A290824 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*x)/(1 + LambertW(-x)).

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 1, 3, 7, 27, 1, 4, 12, 43, 256, 1, 5, 19, 71, 393, 3125, 1, 6, 28, 117, 616, 4721, 46656, 1, 7, 39, 187, 985, 7197, 69853, 823543, 1, 8, 52, 287, 1584, 11123, 105052, 1225757, 16777216, 1, 9, 67, 423, 2521, 17429, 159093, 1829291, 24866481, 387420489, 1, 10, 84, 601, 3928, 27525, 243256, 2740111, 36922928, 572410513, 10000000000

Offset: 0

Ilya Gutkovskiy, Aug 11 2017

A(n,k) is the k-th binomial transform of A000312 evaluated at n.

			E.g.f. of column k: A_k(x) = 1 + (k + 1)*x/1! + (k^2 + 2*k + 4)*x^2/2! + (k^3 + 3*k^2 + 12*k + 27)*x^3/3! + (k^4 + 4*k^3 + 24*k^2 + 108*k + 256)*x^4/4! + ...
Square array begins:
     1,     1,     1,     1,     1,     1, ...
     1,     2,     3,     4,     5,     6, ...
     4,     7,    12,    19,    28,    39, ...
    27,    43,    71,   117,   187,   287, ...
   256,   393,   616,   985,  1584,  2521, ...
  3125,  4721,  7197, 11123, 17429, 27525, ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
N. J. A. Sloane, Transforms

Columns k=0..2 give A000312, A086331, A277457.
Main diagonal gives A290840.
Cf. A080955, A081576, A271025.

Table[Function[k, n!*SeriesCoefficient[Exp[k x]/(1 + LambertW[-x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten (* G. C. Greubel, Nov 09 2017 *)

E.g.f. of column k: exp(k*x)/(1 + LambertW(-x)).

A(n,k) = Sum_{j=0..n} binomial(n,j)*k^(n-j)*j^j. - Fabian Pereyra, Jul 16 2024

A292633 a(n) = n! * [x^n] -exp(nx)LambertW(-x).

Original entry on oeis.org

0, 1, 6, 54, 656, 10100, 189252, 4195870, 107803712, 3158565192, 104179336100, 3827097857594, 155176637687568, 6890781261435916, 332846314505306084, 17384125179840159150, 976545328548757184768, 58723524484105985029136, 3764267361608204263229892, 256245748998712921762125922

Offset: 0

Ilya Gutkovskiy, Sep 20 2017

nonn

The n-th term of the n-th binomial transform of A000169.

G. C. Greubel, Table of n, a(n) for n = 0..365
N. J. A. Sloane, Transforms

Main diagonal of A294411.

Cf. A000169, A290840.

Table[n!*SeriesCoefficient[-E^(n*x)*LambertW[-x],{x,0,n}], {n,0,20}] (* Vaclav Kotesovec, Sep 20 2017 *)

a(n) ~ exp(n*exp(-1)) * n^(n-1) / (1-exp(-1))^(3/2). - Vaclav Kotesovec, Sep 20 2017

cp's OEIS Frontend

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

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A290824 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*x)/(1 + LambertW(-x)).

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A292633 a(n) = n! * [x^n] -exp(nx)LambertW(-x).

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

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Crossrefs

Programs

Mathematica

PARI

Formula

A290824 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*x)/(1 + LambertW(-x)).

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Author

Keywords

Comments

Examples

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Crossrefs

Programs

Mathematica

Formula

A292633 a(n) = n! * [x^n] -exp(n*x)*LambertW(-x).

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Formula

A292633 a(n) = n! * [x^n] -exp(nx)LambertW(-x).