A363864 a(n) = A143007(2*n,n).

1, 13, 661, 46705, 3833941, 342981013, 32443313449, 3191377294153, 323158664855125, 33461619685494025, 3526747995762849661, 377103695616260979313, 40807220545026078297961, 4460530114068960956304865, 491780450707942086338993761, 54624008737670717933342875705

Offset: 0

Author

Peter Bala, Jun 25 2023

Keywords

• nonn
• easy

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.

More generally, for positive integers r and s the sequence {A143007(r*n, s*n) : n >= 0} satisfies the same supercongruences. For cases, see A005259(r = s = 1), A363865 (r = 3, s = 1) and A363866 (r = 3, s = 2).

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.

Crossrefs

Cf. A005259, A143007, A108625, A363865 - A363871.

Programs

Maple

A143007 := proc(n, k); add(binomial(n+j, 2*j)*binomial(2*j, j)^2*binomial(k+j, 2*j), j = 0..n) end:
seq(A143007(2*n, n), n = 0..20);
# alternative program
seq(simplify(hypergeom([2*n+1, -2*n, n+1, -n], [1, 1, 1], 1)), n = 0..20);

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