A290824 -id:A290824 - cp's OEIS frontend

A290840 a(n) = n! * [x^n] exp(n*x)/(1 + LambertW(-x)).

1, 2, 12, 117, 1584, 27525, 585108, 14726411, 428551616, 14161828185, 523952280900, 21456869976135, 963553844335536, 47078974421716757, 2486272976536821332, 141118622400977894475, 8566597074999702384384, 553816179165426157329201, 37985975117322654130568964

Offset: 0

Ilya Gutkovskiy, Aug 12 2017

nonn

G. C. Greubel, Table of n, a(n) for n = 0..365
N. J. A. Sloane, Transforms
Martin Svatoš, Peter Jung, Jan Tóth, Yuyi Wang, and Ondřej Kuželka, On Discovering Interesting Combinatorial Integer Sequences, arXiv:2302.04606 [cs.LO], 2023, p. 17.

Main diagonal of A290824.

Cf. A000312, A207833, A254382.

Table[n! * SeriesCoefficient[Exp[n*x]/(1 + LambertW[-x]), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 06 2017 *)

a(n) = A290824(n,n).
a(n) ~ exp(1/2 + n*exp(-1)) * n^n / sqrt(exp(1)-1). - Vaclav Kotesovec, Oct 06 2017
a(n) = Sum_{k=0..n} binomial(n,k)*n^(n-k)*k^k. - Fabian Pereyra, Jul 16 2024
E.g.f.: 1/((1+LambertW(-x))*(1+LambertW(LambertW(-x)))). - Fabian Pereyra, Jul 19 2024

A294411 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. -exp(kx)LambertW(-x).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 4, 9, 0, 1, 6, 18, 64, 0, 1, 8, 33, 116, 625, 0, 1, 10, 54, 216, 1060, 7776, 0, 1, 12, 81, 388, 1865, 12702, 117649, 0, 1, 14, 114, 656, 3340, 21228, 187810, 2097152, 0, 1, 16, 153, 1044, 5905, 36414, 303765, 3296120, 43046721, 0, 1, 18, 198, 1576, 10100, 63480, 500374, 5222864, 66897288, 1000000000

Offset: 0

Ilya Gutkovskiy, Oct 30 2017

			E.g.f. of column k: A_k(x) = x/1! + 2*(k + 1)*x^2/2! + 3*(k^2 + 2*k + 3)*x^3/3! + 4*(k^3 + 3*k^2 + 9*k + 16)*x^4/4! + ...
Square array begins:
    0,     0,     0,     0,     0,      0, ...
    1,     1,     1,     1,     1,      1, ...
    2,     4,     6,     8,    10,     12, ...
    9,    18,    33,    54,    81,    114, ...
   64,   116,   216,   388,   656,   1044, ...
  625,  1060,  1895,  3340,  5905,  10100, ...

G. C. Greubel, Table of n, a(n) for the first 50 antidiagonals, flattened

Columns k=0..2 give A000169, A277473, A277485.
Main diagonal gives A292633.
Cf. A290824.

Table[Function[k, n! SeriesCoefficient[-Exp[k x] LambertW[-x], {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

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

cp's OEIS Frontend

A290840 a(n) = n! * [x^n] exp(n*x)/(1 + LambertW(-x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A294411 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. -exp(kx)LambertW(-x).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

cp's OEIS Frontend

A290840 a(n) = n! * [x^n] exp(n*x)/(1 + LambertW(-x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A294411 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)*LambertW(-x).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A294411 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. -exp(kx)LambertW(-x).