OFFSET
0,1
FORMULA
a(n) = (6*(n+1)/(3+floor(n/2)))*A056040(n).
a(2*n) = A007946(n).
Recurrence: a(n) = a(n-1)*(n+1)*(4*(n+4))/(n^2*(n+6)) if n mod 2 = 0 else a(n-1)*(n+1) for n>0, a(0) = 2.
Asymptotic: a(x) ~ exp(x*log(2) - log(Pi)/2 - cos(Pi*x)*(log(x/2) + 1/(2*x))/2 + log(6*(x+1)) - log(3+floor(x/2))) for x>=1.
G.f.: (4*x-1)/(2*x^6) + (-16*x^7+16*x^6-48*x^5+12*x^4+48*x^3-12*x^2-8*x+2)/(4*(1-4*x^2)^(3/2)*x^6). - Robert Israel, Aug 25 2014
Sum_{n>=0} 1/a(n) = Pi^2/54 + 19*Pi/(54*sqrt(3)) + 1/9. - Amiram Eldar, Feb 17 2023
MAPLE
MATHEMATICA
Table[6(n + 1)!/((3 + Floor[n/2])*(Floor[n/2]!)^2), {n, 0, 30}] (* Wesley Ivan Hurt, Aug 26 2014 *)
PROG
(Sage)
@CachedFunction
def A242986(n):
if n == 0: return 2
h = (n+1)*A242986(n-1)
if 2.divides(n):
h *= (4*(n+4))/(n^2*(n+6))
return h
[A242986(n) for n in range(33)]
(Magma) [6*Factorial(n+1)/((3+Floor(n/2))*(Factorial(Floor(n/2)))^2) : n in [0..30]]; // Wesley Ivan Hurt, Aug 26 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Luschny, Aug 25 2014
STATUS
approved