OFFSET
0,1
COMMENTS
Row 1 of A361032.
The central binomial numbers A000984(n) = (2*n)!/n!^2 have the property that 6*A000984(n) is divisible by (n + 1)*(n + 2) and the result (2*n)!/(n!*(n+2)!) is the super ballot number A007054(n). Similarly, the numbers A008977(n) = (4*n)!/n!^4 appear to have the property that 2520*A008977(n) is divisible by ((n + 1)*(n + 2))^3, leading to the present sequence. Cf. A361029.
Conjecture: a(n) is odd iff n = 2^k - 2 for some k >= 1.
FORMULA
a(n) = 2520*A008977(n)/((n+1)*(n+2))^3.
a(n) = (315/2)*A008977(n+2)/((4*n+1)*(4*n+2)*(4*n+3)*(4*n+5)*(4*n+6)*(4*n+7)).
P-recursive: a(n) = 4*(4*n-1)*(4*n-2)*(4*n-3)/(n+2)^3 * a(n-1) with a(0) = 315.
The o.g.f. A(x) satisfies the differential equation
x^3*(1 - 256*x)*A(x)''' + x^2*(9 - 1152*x)*A(x)'' + x*(19 - 816*x)*A(x)' + (8 - 24*x)*A(x) - 2520 = 0 with A(0) = 315, A'(0) = 280 and A''(0) = 7350.
a(n) ~ 630*sqrt(8/Pi^3) * 2^(8*n)/n^(15/2).
MAPLE
seq(2520*(4*n)!/(n!*(n+2)!^3), n = 0..20);
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Mar 01 2023
STATUS
approved