%I #9 Jun 30 2023 08:18:47
%S 1,253,494341,1403375905,4684608730309,17126002734202253,
%T 66366682204430084569,267832273159817887638881,
%U 1113652383352571992799711941,4737943697041408831021629805753,20526206833382185439454748049996341
%N a(n) = A143007(3*n,2*n).
%C 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.
%C More generally, for positive integers r and s, the sequence {A143007(r*n, s*n) : n >= 0} satisfies the above supercongruences. For other cases, see A005259 (r = s = 1), A363864 (r = 2, s = 1) and A363865 (r = 3, s = 1).
%H Peter Bala, <a href="/A363866/a363866_1.pdf">A recurrence for A363866</a>
%H Armin Straub, <a href="http://dx.doi.org/10.2140/ant.2014.8.1985">Multivariate Apéry numbers and supercongruences of rational functions</a>, Algebra & Number Theory, Vol. 8, No. 8 (2014), pp. 1985-2008; <a href="https://arxiv.org/abs/1401.0854">arXiv preprint</a>, arXiv:1401.0854 [math.NT], 2014.
%F a(n) = Sum_{k = 0..2*n} binomial(3*n+k,2*k)*binomial(2*k,k)^2*binomial(2*n+k,2*k).
%F a(n) = Sum_{k = 0..2*n} binomial(3*n,k)^2*binomial(5*n-k,2*n-k)^2.
%F a(n) = hypergeom([3*n+1, -2*n, 3*n+1, -2*n], [1, 1, 1], 1)
%F 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.
%p A143007 := proc(n, k); add(binomial(n+j, 2*j)*binomial(2*j, j)^2*binomial(k+j, 2*j), j = 0..n) end:
%p seq(A143007(3*n, 2*n), n = 0..20);
%p # alternative program
%p seq(simplify(hypergeom([3*n+1, -3*n, 2*n+1, -2*n], [1, 1, 1], 1)), n = 0..20);
%Y Cf. A005259, A143007, A363864 - A363871.
%K nonn,easy
%O 0,2
%A _Peter Bala_, Jun 25 2023