A274279 Expansion of e.g.f.: tanh(x*W(x)), where W(x) = LambertW(-x)/(-x).
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
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
- Paul D. Hanna, Table of n, a(n) for n = 1..200
Programs
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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