|
Search: id:A116637
|
|
|
| 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. |
|
+0
1
|
|
| 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
(list; graph; listen)
|
|
|
OFFSET
|
0,2
|
|
|
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 (pbarry(AT)wit.ie), Jul 07 2007
|
|
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 +...
|
|
PROGRAM
|
(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)!)))}
(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
|
Cf. A046646, A001764, A006013.
Sequence in context: A074028 A061894 A116684 this_sequence A134041 A069925 A080611
Adjacent sequences: A116634 A116635 A116636 this_sequence A116638 A116639 A116640
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Paul D. Hanna (pauldhanna(AT)juno.com), Feb 19 2006
|
|
|
Search completed in 0.002 seconds
|
