A277461 -id:A277461

A277463

E.g.f.: sinh(x)/(1+LambertW(-x)).

+10
5

0, 1, 2, 13, 112, 1321, 19296, 335637, 6764864, 154946449, 3973820800, 112789880413, 3509627281920, 118790978349369, 4344883388878592, 170767066282574821, 7177162988688031744, 321206181612447781921, 15250250261039350358016, 765586309042945067185581

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

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

FORMULA

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

MATHEMATICA

CoefficientList[Series[Sinh[x]/(1+LambertW[-x]), {x, 0, 25}], x] * Range[0, 25]!

Table[(1-(-1)^n + Sum[(1-(-1)^(n-k)) * Binomial[n, k] * k^k, {k, 1, n}])/2, {n, 0, 25}]

PROG

(PARI) x='x+O('x^50); concat([0], Vec(serlaplace(sinh(x)/(1 + lambertw(-x))))) \\ G. C. Greubel, Nov 05 2017

CROSSREFS

Cf. A000312, A069856, A086331, A277461, A277464.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Oct 16 2016

STATUS

approved

A277462

E.g.f.: cos(x)/(1 + LambertW(-x)).

+10
4

1, 1, 3, 24, 233, 2860, 42875, 758856, 15488657, 358164432, 9254769459, 264273873600, 8264362186489, 280896392748608, 10310601442639147, 406479520869636480, 17129450693008029729, 768404013933189112064, 36557893891263190204259, 1838650651518153170939904

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

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

FORMULA

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

MATHEMATICA

CoefficientList[Series[Cos[x]/(1+LambertW[-x]), {x, 0, 25}], x] * Range[0, 25]!

Table[Cos[Pi*n/2] + Sum[Binomial[n, k] * Cos[Pi*(n-k)/2] * k^k, {k, 1, n}], {n, 0, 25}] (* Vaclav Kotesovec, Oct 28 2016 *)

PROG

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

CROSSREFS

Cf. A000312, A086331, A277461, A277464.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Oct 16 2016

STATUS

approved

A277475

E.g.f.: -sin(x)*LambertW(-x).

+10
4

0, 0, 2, 6, 32, 300, 3576, 52234, 906688, 18229176, 416505760, 10657541422, 301871501568, 9375794555556, 316817746172032, 11570642333807730, 454152692297009152, 19064517871187079408, 852278820775206658560, 40424330665968520135382, 2027524052626732381306880

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

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

FORMULA

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

MATHEMATICA

CoefficientList[Series[-Sin[x]*LambertW[-x], {x, 0, 20}], x] * Range[0, 20]!

Table[Sum[Binomial[n, k] * Sin[Pi*(n-k)/2] * k^(k-1), {k, 1, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 28 2016 *)

PROG

(PARI) x='x+O('x^50); concat([0, 0], Vec(serlaplace(-sin(x)*lambertw(-x)) )) \\ G. C. Greubel, Nov 07 2017

CROSSREFS

Cf. A000169, A277461, A277473, A277476, A277477.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Oct 17 2016

STATUS

approved

A277498

E.g.f.: -LambertW(-sin(x)).

+10
3

0, 1, 2, 8, 56, 536, 6528, 96592, 1683072, 33760576, 766283264, 19417068032, 543351873536, 16642224306176, 553782090473472, 19893884376859648, 767355755629215744, 31631864049541107712, 1387750771948607504384, 64561526675221208563712

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

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

FORMULA

a(n) ~ (1-exp(-2))^(1/4) * (arcsin(exp(-1)))^(1/2-n) * exp(1/2-n) * n^(n-1).

MATHEMATICA

CoefficientList[Series[-LambertW[-Sin[x]], {x, 0, 20}], x] * Range[0, 20]!

PROG

(PARI) x='x+O('x^50); concat([0], Vec(serlaplace(- lambertw(-sin(x))))) \\ G. C. Greubel, Nov 08 2017

CROSSREFS

Cf. A238085, A277461, A277475, A277499, A277500.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Oct 18 2016

STATUS

approved

A277499

E.g.f.: -sin(LambertW(-x)).

+10
2

0, 1, 2, 8, 52, 476, 5646, 82368, 1426888, 28623376, 652516090, 16660233600, 470930272572, 14598765522368, 492441140292934, 17955574113204224, 703714660937658128, 29500170665998713088, 1317136516654501334898, 62399954043306802391040

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Robert Israel, Table of n, a(n) for n = 0..387

FORMULA

a(n) ~ cos(1) * n^(n-1).

MAPLE

S:= series(-sin(LambertW(-x)), x, 31):

seq(coeff(S, x, n)*n!, n=0..30); # Robert Israel, Oct 30 2016

MATHEMATICA

CoefficientList[Series[-Sin[LambertW[-x]], {x, 0, 20}], x] * Range[0, 20]!

PROG

(PARI) x='x+O('x^50); concat([0], Vec(serlaplace(-sin(lambertw(-x))))) \\ G. C. Greubel, Nov 08 2017

CROSSREFS

Cf. A195136, A277461, A277498.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Oct 18 2016

STATUS

approved