A295255 Expansion of e.g.f. 2/(1 + sqrt(1 - 4*x*cos(x))).

1, 1, 4, 27, 288, 4145, 75360, 1655003, 42628096, 1260274689, 42070233600, 1565308844539, 64237925148672, 2882670856605553, 140430196702035968, 7380867094885024635, 416320345406371921920, 25084955259883686000257, 1608058868442709001895936, 109278344982307590211482971

Offset: 0

Author

Ilya Gutkovskiy, Nov 18 2017

Keywords

• nonn

Crossrefs

Cf. A000108, A295237, A295254, A295256, A295257, A295258.

Programs

Maple

a:=series(2/(1+sqrt(1-4*x*cos(x))),x=0,21): seq(n!*coeff(a,x,n),n=0..19); # Paolo P. Lava, Mar 27 2019

Mathematica

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

Formula

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

a(n) ~ sqrt(2 - 2*r*sqrt(16*r^2 - 1)) * n^(n-1) / (exp(n) * r^n), where r = A196605 = 0.2585985822541894903... is the root of the equation r*cos(r) = 1/4. - Vaclav Kotesovec, Nov 18 2017