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%I #28 Mar 12 2024 02:41:45
%S 1,2,7,40,341,3936,57107,992384,20025385,459466240,11804134079,
%T 335571265536,10456512176189,354362575314944,12975301760361163,
%U 510462668072058880,21472710312090391889,961728814178702327808,45692671937666739799799,2295278998002033651875840,121545436687537993689631525,6767130413049423041105231872,395177438856180565803457658627,24152146710231984411570685870080
%N Expansion of e.g.f.: tanh(x*W(x)), where W(x) = LambertW(-x)/(-x).
%H Paul D. Hanna, <a href="/A274279/b274279.txt">Table of n, a(n) for n = 1..200</a>
%F 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!.
%F a(n) ~ 4*exp(2) * n^(n-1) / (1+exp(2))^2. - _Vaclav Kotesovec_, Jun 23 2016
%F a(n) = Sum_{k=0..n-1} (-1)^k * A264902(n,k). - _Alois P. Heinz_, Aug 08 2022
%e 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! +...
%e such that A(x) = tanh(x*W(x))
%e where W(x) = LambertW(-x)/(-x) begins
%e 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! +...
%e and satisfies W(x) = exp(x*W(x)).
%e Also, A(x) = (W(x)^2 - 1)/(W(x)^2 + 1), where
%e 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! +...
%t 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 *)
%t Rest[With[{nmax=30}, CoefficientList[Series[Tanh[-LambertW[-x]], {x,0,nmax}], x]*Range[0, nmax]!]] (* _G. C. Greubel_, Feb 19 2018 *)
%o (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)}
%o for(n=1, 30, print1(a(n), ", "))
%o (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)}
%o for(n=1, 30, print1(a(n), ", "))
%o (PARI) x='x+O('x^30); Vec(serlaplace(tanh(-lambertw(-x)))) \\ _G. C. Greubel_, Feb 19 2018
%Y Cf. A000272, A195136, A238085, A264902, A274278, A277468, A277480.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jun 19 2016
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