A166748 E.g.f.: exp(6*arcsin(x)).

1, 6, 36, 222, 1440, 9990, 74880, 609390, 5391360, 51798150, 539136000, 6060383550, 73322496000, 951480217350, 13198049280000, 195053444556750, 3061947432960000, 50908949029311750, 894088650424320000

Offset: 0

Author

Jaume Oliver Lafont, Oct 21 2009

Keywords

• nonn

Comments

exp(6*arcsin(1/2)) is Aleksandr Gelfond's constant exp(Pi).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..445
• A. R. Povolotsky et al., With regards to OEIS A166748, sci.math.symbolic usenet group, 2009

Crossrefs

Cf. A166741, A006228, A039661.

Programs

Mathematica

Round[Table[3*2^(n-1)*(E^(3*Pi)-(-1)^n*E^(-3*Pi))*Abs[Gamma[n/2+3*I]]^2/Pi,{n,0,20}]] (* Vaclav Kotesovec, Nov 06 2014 *)
CoefficientList[Series[Exp[6*ArcSin[x]], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Nov 06 2014 *)

PARI

A166748(n)=round(norm(gamma(n/2+3*I))/Pi*if(n%2,cosh(3*Pi),sinh(3*Pi))*3<M. F. Hasler, Oct 25 2009

PARI

a(n)=polcoeff(exp(6*asin(x)),n)*n!

PARI

a(n)=(1+5*(n%2))*prod(k=0,n\2-1,(2*k+n%2)^2+36) \\ Jaume Oliver Lafont, Oct 28 2009

Formula

Contribution from Alexander R. Povolotsky, Oct 24 2009: (Start)

a(n+2) = (n^2+36)*a(n), a(0)=1, a(1)=6.

The above recurrence leads to

a(n) = (3*2^n*gamma(-3*i+n/2)*gamma(3*i+n/2)*(cos((n*Pi)/2)+i*sin((n*Pi)/2))*sinh(((6-i*n)*Pi)/2))/Pi where "i" is imaginary unit. (End)

a(n) = 3*2^(n-1)*(exp(3*Pi)-(-1)^n*exp(-3*Pi))*|Gamma(n/2+3i)|^2/Pi. - R. J. Mathar and M. F. Hasler, Oct 25 2009

a(n) ~ 6 * (exp(3*Pi) - (-1)^n*exp(-3*Pi)) * n^(n-1) / exp(n). - Vaclav Kotesovec, Nov 06 2014

Extensions

Minor edits by Vaclav Kotesovec, Nov 06 2014