OFFSET
0,2
COMMENTS
a(n) = A(3*n,2*n,3*n,2*n) in the notation of Straub, equation 8. It follows from Straub, Theorem 1.2, that the supercongruence a(n*p^k) == a(n*p^(k-1)) (mod p^(3*k)) holds for all primes p >= 5 and all positive integers n and k.
LINKS
Peter Bala, A recurrence for A363866
Armin Straub, Multivariate Apéry numbers and supercongruences of rational functions, Algebra & Number Theory, Vol. 8, No. 8 (2014), pp. 1985-2008; arXiv preprint, arXiv:1401.0854 [math.NT], 2014.
FORMULA
a(n) = Sum_{k = 0..2*n} binomial(3*n+k,2*k)*binomial(2*k,k)^2*binomial(2*n+k,2*k).
a(n) = Sum_{k = 0..2*n} binomial(3*n,k)^2*binomial(5*n-k,2*n-k)^2.
a(n) = hypergeom([3*n+1, -2*n, 3*n+1, -2*n], [1, 1, 1], 1)
a(n) = [x^(2*n)] 1/(1 - x)*( Legendre_P(3*n,(1 + x)/(1 - x)) )^2 = [x^(3*n)] 1/(1 - x)*( Legendre_P(2*n,(1 + x)/(1 - x)) )^2.
MAPLE
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Jun 25 2023
STATUS
approved