OFFSET
0,2
COMMENTS
Row 4 of A364513.
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..400
FORMULA
a(n) = [x^n] (1 - x)^(2*n) * Legendre_P(4*n-1, (1 + x)/(1 - x)) for n >= 1.
a(n) = Sum_{k = 0..n} binomial(4*n-1, n-k)^2 * binomial(2*n+k-2, k).
a(n) = (1/2) * binomial(6*n, 2*n)*binomial(2*n, n)^2 / binomial(3*n, n).
a(n) = (4*n - 1)!*(3*n - 1/2)!*(n - 1/2)!/((2*n - 1)! * (2*n - 1/2)!^2 * n!^2) for n >= 1 (fractional factorials are defined in terms of the gamma function, for example, (3*n - 1/2)! = gamma(3*n + 1/2)).
a(n) ~ 108^n * sqrt(2)/(4*Pi*n).
P-recursive: a(0) = 1; for n >= 1, a(n) = 12*(6*n-1)*(6*n-5)*(2*n-1)^2 / ((4*n-1)*(4*n-3)*n^2) * a(n-1) with a(1) = 10.
Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and all positive integers n and r.
EXAMPLE
Examples of supercongruences:
a(7) - a(1) = 2677247862432 - 10 = 2*(7^3)*1999*1952323 == 0 (mod 7^3).
a(2*7) - a(2) = 232736024465276636260692000 - 594 = 2*(3^3)*(7^3)*1097* 11454314520615738059 == 0 (mod 7^3).
MAPLE
seq( (1/2) * (6*n)!*(2*n)!^2/((3*n)!*(4*n)!*n!^3), n = 1..15);
MATHEMATICA
A364515[n_]:=If[n==0, 1, (1/2)(6n)!(2n)!^2/((3n)!(4n)!n!^3)]; Array[A364515, 15, 0] (* Paolo Xausa, Oct 05 2023 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Aug 01 2023
STATUS
approved