A277485 -id:A277485 - cp's OEIS frontend

A277457 E.g.f.: exp(2*x)/(1+LambertW(-x)).

1, 3, 12, 71, 616, 7197, 105052, 1829291, 36922928, 846851993, 21744781684, 617832652527, 19242299657896, 651815827343189, 23857403245171724, 938247816632341043, 39455261828928309088, 1766645684585351990961, 83913998998426051745764, 4214295288128637488870327, 223120214856875472660345176

Offset: 0

Vaclav Kotesovec, Oct 16 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A086331, A277454, A277456, A277485.

CoefficientList[Series[Exp[2*x]/(1+LambertW[-x]), {x, 0, 20}], x]*Range[0, 20]!
Table[1 + Sum[Binomial[n, m]*(1 + Sum[Binomial[m, k]*k^k, {k, 1, m}]), {m, 1, n}], {n, 0, 20}]
Table[2^n + Sum[Binomial[n, k]*2^(n-k)*k^k, {k, 1, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 28 2016 *)

x='x+O('x^50); Vec(serlaplace(exp(2*x)/(1 + lambertw(-x)))) \\ G. C. Greubel, Nov 07 2017

a(n) ~ exp(2*exp(-1)) * n^n.

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

A277457 E.g.f.: exp(2*x)/(1+LambertW(-x)).

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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).

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cp's OEIS Frontend

A277457 E.g.f.: exp(2*x)/(1+LambertW(-x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

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).