A295257 Expansion of e.g.f. cot(x)*(1 - sqrt(1 - 4*tan(x)))/2.

1, 1, 4, 32, 368, 5656, 109024, 2533712, 68995328, 2155513216, 76014982144, 2987332904192, 129473128921088, 6135478762187776, 315609465774936064, 17515027337549545472, 1043104219010147483648, 66358462250378681614336, 4491141928841064201846784, 322219449242531127348887552

Offset: 0

Author

Ilya Gutkovskiy, Nov 18 2017

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..365

Crossrefs

Cf. A000108, A195727, A295237, A295254, A295255, A295256, A295258.

Programs

Maple

S:= series(cot(x)*(1 - sqrt(1 - 4*tan(x)))/2, x, 32):
seq(n!*coeff(S,x,n),n=0..30); # Robert Israel, Nov 18 2017

Mathematica

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

Formula

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

a(n) ~ sqrt(17/2) * n^(n-1) / (exp(n) * (arctan(1/4))^(n-1/2)). - Vaclav Kotesovec, Nov 18 2017