OFFSET
0,2
COMMENTS
a(n) = A(2*n,n,2*n,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
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..n} binomial(2*n,n-k)^2*binomial(2*n+k,k)^2.
a(n) = Sum_{k = 0..n} binomial(2*n+k,2*k)*binomial(2*k,k)^2*binomial(n+k,2*k).
a(n) = hypergeom([2*n+1, -2*n, n+1, -n], [1, 1, 1], 1)
a(n) = [x^n] 1/(1 - x)*( Legendre_P(2*n,(1 + x)/(1 - x)) )^2 = [x^(2*n)] 1/(1 - x)*( Legendre_P(n,(1 + x)/(1 - x)) )^2.
P-recursive: 2*(440*n^3 - 1782*n^2 + 2412*n - 1091)*(2*n - 1)^3*n^3*a(n) = (865920*n^9 - 6104736*n^8 + 18475432*n^7 - 31464562*n^6 + 33227280*n^5 - 22586875*n^4 + 9902182*n^3 - 2707173*n^2 + 420336*n - 28350)*a(n-1) - 2*(440*n^3 - 462*n^2 + 168*n - 21)*(n - 1)^3*(2*n - 3)^3*a(n-2) with a(0) = 1 and a(1) = 13.
a(n) = Sum_{k = 0..2*n} (-1)^k*binomial(2*n, k)*binomial(2*n+k, k)*A108625(n, k) (verified using the MulZeil procedure in Doron Zeilberger's MultiZeilberger package). - Peter Bala, Oct 16 2024
MAPLE
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Jun 25 2023
STATUS
approved