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Expansion of e.g.f. csch(x)*(1 - sqrt(1 - 4*sinh(x)))/2.
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%I #9 Mar 27 2019 10:02:11

%S 1,1,4,31,352,5341,101824,2341291,63092992,1950837241,68093599744,

%T 2648776394551,113633946898432,5330308817264341,271416230974603264,

%U 14910196369733535811,879003840976919068672,55354496206857969062641,3708594029795800700944384,263391744037123969891925071

%N Expansion of e.g.f. csch(x)*(1 - sqrt(1 - 4*sinh(x)))/2.

%F E.g.f.: 1/(1 - sinh(x)/(1 - sinh(x)/(1 - sinh(x)/(1 - sinh(x)/(1 - ...))))), a continued fraction.

%F a(n) ~ sqrt(2) * 17^(1/4) * n^(n-1) / (exp(n) * (log((1+ sqrt(17))/4))^(n - 1/2)). - _Vaclav Kotesovec_, Nov 18 2017

%p a:=series(csch(x)*(1-sqrt(1-4*sinh(x)))/2,x=0,21): seq(n!*coeff(a,x,n),n=0..19); # _Paolo P. Lava_, Mar 27 2019

%t nmax = 19; CoefficientList[Series[Csch[x] (1 - Sqrt[1 - 4 Sinh[x]])/2, {x, 0, nmax}], x] Range[0, nmax]!

%t nmax = 19; CoefficientList[Series[1/(1 + ContinuedFractionK[-Sinh[x], 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

%Y Cf. A000108, A295237, A295255, A295256, A295257, A295258.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Nov 18 2017