|
|
A166748
|
|
E.g.f.: exp(6*arcsin(x)).
|
|
4
|
|
|
1, 6, 36, 222, 1440, 9990, 74880, 609390, 5391360, 51798150, 539136000, 6060383550, 73322496000, 951480217350, 13198049280000, 195053444556750, 3061947432960000, 50908949029311750, 894088650424320000
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
exp(6*arcsin(1/2)) is Aleksandr Gelfond's constant exp(Pi).
|
|
LINKS
|
|
|
FORMULA
|
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) ~ 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!
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|