A166741 E.g.f.: exp(2*arcsin(x)).

1, 2, 4, 10, 32, 130, 640, 3770, 25600, 199810, 1740800, 16983850, 181043200, 2122981250, 26794393600, 367275756250, 5358878720000, 84106148181250, 1393308467200000, 24643101417106250, 457005177241600000

Offset: 0

Author

Jaume Oliver Lafont, Oct 21 2009

Keywords

• nonn

Comments

exp(2*arcsin(1)) is Aleksandr Gelfond's constant.

Links

• Wikipedia, Gelfond's constant

Crossrefs

Cf. A006228, A039661, A166748.

Programs

Maple

seq(simplify(2^(n-1) * (cosh(Pi)*(1-(-1)^n) + sinh(Pi)*(1+(-1)^n)) * GAMMA((1/2)*n-I)*GAMMA((1/2)*n+I) / Pi), n=0..20); # Vaclav Kotesovec, Nov 06 2014

Mathematica

CoefficientList[Series[E^(2*ArcSin[x]), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Aug 04 2014 *)
FullSimplify[Table[2^(n-1) * (E^(Pi)-(-1)^n*E^(-Pi)) * Gamma[n/2-I] * Gamma[n/2+I] / Pi,{n,0,20}]] (* Vaclav Kotesovec, Nov 06 2014 *)

PARI

for (n=0,25,print(polcoeff(exp(2*asin(x)),n)*n!,","))

Formula

a(n) ~ 2 * n^(n-1) * (exp(Pi) - (-1)^n/exp(Pi)) / exp(n). - Vaclav Kotesovec, Aug 04 2014

From Vaclav Kotesovec, Nov 06 2014: (Start)

a(n) = (n^2 - 4*n + 8)*a(n-2).

a(n) = 2^(n-1) * (exp(Pi)-(-1)^n*exp(-Pi)) * GAMMA(n/2-I) * GAMMA(n/2+I) / Pi.

(End)