A295237 Expansion of e.g.f. csc(x)*(1 - sqrt(1 - 4*sin(x)))/2.

1, 1, 4, 29, 320, 4741, 88384, 1988489, 52448000, 1587545161, 54252120064, 2066298252149, 86799115489280, 3986897970744781, 198795278022098944, 10694247962623751009, 617392620634705756160, 38074395493710549747601, 2498063366053169206657024, 173745719989547715852773069

Offset: 0

Author

Ilya Gutkovskiy, Nov 18 2017

Keywords

• nonn

Crossrefs

Cf. A000108, A195621, A295254, A295255, A295256, A295257, A295258.

Programs

Maple

a:=series(csc(x)*(1-sqrt(1-4*sin(x)))/2,x=0,20): seq(n!*coeff(a,x,n),n=0..19); # Paolo P. Lava, Mar 27 2019

Mathematica

nmax = 19; CoefficientList[Series[Csc[x] (1 - Sqrt[1 - 4 Sin[x]])/2, {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[1/(1 + ContinuedFractionK[-Sin[x], 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: 1/(1 - sin(x)/(1 - sin(x)/(1 - sin(x)/(1 - sin(x)/(1 - ...))))), a continued fraction.

a(n) ~ sqrt(2) * 15^(1/4) * n^(n-1) / (exp(n) * (arcsin(1/4))^(n - 1/2)). - Vaclav Kotesovec, Nov 18 2017