OFFSET
1,4
COMMENTS
Part (ii) of the conjecture in A246065 implies that all the terms in the current sequence are integers.
Conjecture: The sequence a(n+1)/a(n) (n = 4,5,...) is strictly increasing to the limit 9, and the sequence a(n+1)^(1/(n+1))/a(n)^(1/n) (n = 3,4,...) is strictly decreasing to the limit 1.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..170
FORMULA
Recurrence: n^2*a(n) = 2*(n-2)*(5*n-8)*a(n-1) - 9*(n-2)^2*a(n-2). - Vaclav Kotesovec, Aug 27 2014
a(n) ~ 3^(2*n+5/2) / (128*Pi*n^4). - Vaclav Kotesovec, Aug 27 2014
EXAMPLE
a(5) = 9 since sum_{k=0}^{5-1}A246065(k) = -1 + 1 + 9 + 39 + 177 = 225 = 5^2*9.
MAPLE
ogf := (1-((9*x-1)/(x-1))^(3/4)*hypergeom([-1/4, 3/4], [1], -64*x/(9*x-1)^3/(x-1)))/6;
series(ogf, x=0, 25); # Mark van Hoeij, Nov 12 2023
MATHEMATICA
s[n_]:=Sum[Binomial[n, k]^2*Binomial[2k, k]/(2k-1), {k, 0, n}]
a[n_]:=Sum[s[k], {k, 0, n-1}]/n^2
Table[a[n], {n, 1, 25}]
CROSSREFS
KEYWORD
sign
AUTHOR
Zhi-Wei Sun, Aug 25 2014
STATUS
approved