A116637 G.f. satisfies: A(x) = x/series_reversion(x/G(x)) where A(x) + A(-x) = 2*G(x^2) and G(x) is the g.f. of A046646.

1, 2, 2, 4, 6, 14, 24, 60, 110, 286, 546, 1456, 2856, 7752, 15504, 42636, 86526, 240350, 493350, 1381380, 2861430, 8064030, 16829280, 47682960, 100134216, 284997384, 601661144, 1719031840, 3645533040, 10450528048, 22249511328, 63967345068

Offset: 0

Author

Paul D. Hanna, Feb 19 2006

Keywords

• nonn

Examples

			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 +...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A046646, A001764, A006013.

Programs

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 *)

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 */

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