cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A219503 Expansion of e.g.f. Sum_{n>=0} (n+1)^(n-1) * sinh(x)^n / n!.

Original entry on oeis.org

1, 1, 3, 17, 137, 1457, 19355, 308961, 5766353, 123285153, 2972114803, 79782059249, 2360417058521, 76319622510289, 2677629295171979, 101318751122847297, 4113158120834726049, 178328823993199602241, 8223999403291995520995, 401989145900847087408849
Offset: 0

Views

Author

Paul D. Hanna, Nov 20 2012

Keywords

Comments

Compare to the LambertW identity: Sum_{n>=0} (n+1)^(n-1)*exp(-n*x)*x^n/n! = exp(x).

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 137*x^4/4! + 1457*x^5/5! +...
where
A(x) = 1 + sinh(x) + 3^1*sinh(x)^2/2! + 4^2*sinh(x)^3/3! + 5^3*sinh(x)^4/4! +...

Links

Crossrefs

Cf. A003724, A136630, A162650.
Cf. A238085.

Programs

  • Mathematica
    CoefficientList[Series[-LambertW[-Sinh[x]]/Sinh[x], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 08 2013 *)
  • PARI
    {a(n)=n!*polcoeff(sum(k=0,n,(k+1)^(k-1)*sinh(x + x*O(x^n))^k/k!),n)}
for(n=0,25,print1(a(n),", "))

Formula

E.g.f.: LambertW(-sinh(x)) / (-sinh(x)).
a(n) ~ (1+exp(2))^(1/4) * n^(n-1) / (exp(n-1) * log(exp(-1) +sqrt(1+exp(-2)))^(n-1/2)). - Vaclav Kotesovec, Jul 08 2013
a(n) = Sum_{k=0..n} (k+1)^(k-1) * A136630(n,k). - Seiichi Manyama, Feb 15 2025

A274279 Expansion of e.g.f.: tanh(x*W(x)), where W(x) = LambertW(-x)/(-x).

Original entry on oeis.org

1, 2, 7, 40, 341, 3936, 57107, 992384, 20025385, 459466240, 11804134079, 335571265536, 10456512176189, 354362575314944, 12975301760361163, 510462668072058880, 21472710312090391889, 961728814178702327808, 45692671937666739799799, 2295278998002033651875840, 121545436687537993689631525, 6767130413049423041105231872, 395177438856180565803457658627, 24152146710231984411570685870080
Offset: 1

Views

Author

Paul D. Hanna, Jun 19 2016

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 7*x^3/3! + 40*x^4/4! + 341*x^5/5! + 3936*x^6/6! + 57107*x^7/7! + 992384*x^8/8! + 20025385*x^9/9! + 459466240*x^10/10! + 11804134079*x^11/11! + 335571265536*x^12/12! +...
such that A(x) = tanh(x*W(x))
where W(x) = LambertW(-x)/(-x) begins
W(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! + 16807*x^6/6! + 262144*x^7/7! + 4782969*x^8/8! + 100000000*x^9/9! +...+ (n+1)^(n-1)*x^n/n! +...
and satisfies W(x) = exp(x*W(x)).
Also, A(x) = (W(x)^2 - 1)/(W(x)^2 + 1), where
W(x)^2 = 1 + 2*x + 8*x^2/2! + 50*x^3/3! + 432*x^4/4! + 4802*x^5/5! + 65536*x^6/6! + 1062882*x^7/7! + 20000000*x^8/8! +...+ 2*(n+2)^(n-1)*x^n/n! +...

Links

Crossrefs

Cf. A000272, A195136, A238085, A264902, A274278, A277468, A277480.

Programs

  • Mathematica
    Rest[CoefficientList[Series[(LambertW[-x]^2 - x^2)/(LambertW[-x]^2 + x^2), {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jun 23 2016 *)
Rest[With[{nmax=30}, CoefficientList[Series[Tanh[-LambertW[-x]], {x,0,nmax}], x]*Range[0, nmax]!]] (* G. C. Greubel, Feb 19 2018 *)
  • PARI
    {a(n) = my(W=sum(m=0, n, (m+1)^(m-1)*x^m/m!) +x*O(x^n)); n!*polcoeff(tanh(x*W), n)}
for(n=1, 30, print1(a(n), ", "))
  • PARI
    {a(n) = my(W = sum(m=0, n, (m+1)^(m-1)*x^m/m!) +x*O(x^n)); n!*polcoeff( (W^2 - 1)/(W^2 + 1), n)}
for(n=1, 30, print1(a(n), ", "))
  • PARI
    x='x+O('x^30); Vec(serlaplace(tanh(-lambertw(-x)))) \\ G. C. Greubel, Feb 19 2018

Formula

E.g.f.: (W(x)^2 - 1)/(W(x)^2 + 1), where W(x) = LambertW(-x)/(-x) = exp(x*W(x)) = Sum_{n>=0} (n+1)^(n-1)*x^n/n!.
a(n) ~ 4*exp(2) * n^(n-1) / (1+exp(2))^2. - Vaclav Kotesovec, Jun 23 2016
a(n) = Sum_{k=0..n-1} (-1)^k * A264902(n,k). - Alois P. Heinz, Aug 08 2022

A277498 E.g.f.: -LambertW(-sin(x)).

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Oct 18 2016

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    CoefficientList[Series[-LambertW[-Sin[x]], {x, 0, 20}], x] * Range[0, 20]!
  • PARI
    x='x+O('x^50); concat([0], Vec(serlaplace(- lambertw(-sin(x))))) \\ G. C. Greubel, Nov 08 2017

Formula

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

A277501 E.g.f.: -LambertW(-tanh(x)).

Original entry on oeis.org

0, 1, 2, 7, 48, 461, 5488, 79171, 1347328, 26396185, 585025024, 14473813311, 395433660416, 11824374817893, 384118189803520, 13470784014801787, 507233444671848448, 20411081546839908401, 874130806090067607552, 39696948293418345150327
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 18 2016

Keywords

Links

Crossrefs

Cf. A238085, A277468, A277480, A277500.

Programs

  • Mathematica
    CoefficientList[Series[-LambertW[-Tanh[x]], {x, 0, 20}], x] * Range[0, 20]!
  • PARI
    x='x+O('x^10); concat([0], Vec(serlaplace(-lambertw(-tanh(x))))) \\ G. C. Greubel, Nov 09 2017

Formula

a(n) ~ sqrt(1-exp(-2)) * 2^(n-1/2) * (log((exp(1)+1)/(exp(1)-1)))^(1/2-n) * exp(1/2-n) * n^(n-1).
Showing 1-4 of 4 results.

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