OFFSET
0,2
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
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
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 *)
PROG
(PARI) A166748(n)=round(norm(gamma(n/2+3*I))/Pi*if(n%2, cosh(3*Pi), sinh(3*Pi))*3<<n) \\ 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
CROSSREFS
KEYWORD
nonn
AUTHOR
Jaume Oliver Lafont, Oct 21 2009
EXTENSIONS
Minor edits by Vaclav Kotesovec, Nov 06 2014
STATUS
approved