OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
a(2*n+1) = 2*(3*n+1)!/((n+1)!*(2*n+1)!) = 2*A006013(n), with a(0)=1 and a(2*n+2) = 2*(3*n+3)!/((n+1)!*(2*n+3)!) = 2*A001764(n+1).
G.f. satisfies: A(x) = G(x/A(x)) and A(x*G(x)) = G(x), where G(x) is the g.f. of A046646.
G.f. satisfies: A(x) = 1/A(-x) since log(A(x)) = Sum_{n>=0} 2*A006013(n)*(n+1)/(2n+1)*x^(2n+1) is an odd function.
G.f.: (1+v)/(1-v) where v=2*sqrt(3)*sin(asin(3*sqrt(3)*x/2)/3)/3. - Paul Barry, Jul 07 2007
Conjecture: 4*n*(n+1)*(3*n-1)*a(n) -36*n*a(n-1) -3*(3*n-5)*(3*n+2)*(3*n-4)*a(n-2)=0. - R. J. Mathar, Jun 22 2016
EXAMPLE
A(x) = 1 + 2*x + 2*x^2 + 4*x^3 + 6*x^4 + 14*x^5 + 24*x^6 + 60*x^7 +...
log(A(x)) = 1*2*x + 2*4/3*x^3 + 7*6/5*x^5 + 30*8/7*x^7 + 143*10/9*x^9 +...
MATHEMATICA
k := Floor[(n - 1)/2]; Table[If[n == 0, 1, If[Mod[n, 2] == 1, 2*(3*k + 1)!/((k + 1)!*(2*k + 1)!), 2*(3*k + 3)!/((k + 1)!*(2*k + 3)!)]], {n, 0, 50}] (* G. C. Greubel, Nov 21 2017 *)
PROG
(PARI) {a(n)=local(k=(n-1)\2); if(n==0, 1, if(n%2==1, 2*(3*k+1)!/((k+1)!*(2*k+1)!), 2*(3*k+3)!/((k+1)!*(2*k+3)!)))}
for(n=0, 40, print1(a(n), ", "))
(PARI) {a(n)=if(n<1, n==0, 2*(n+n\2)!/ (n\2+n%2)!/ (n+1-(n%2))!)} /* Michael Somos, Feb 22 2006 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 19 2006
STATUS
approved