cp's OEIS Frontend

A363865 a(n) = A143007(3*n,n).

%I A363865 #10 Jun 30 2023 08:18:42
%S A363865 1,25,2773,430081,77620661,15276834025,3180268712125,688612022804773,
%T A363865 153504724110658741,34994264014955310181,8120680975872203708773,
%U A363865 1911897036160037674700065,455553725980571500127902109
%N A363865 a(n) = A143007(3*n,n).
%C A363865 a(n) = A(3*n,n,3*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.
%C A363865 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 A363866 (r = 3, s = 2).
%H A363865 Peter Bala, <a href="/A363865/a363865.pdf">A recurrence for A363865</a>
%H A363865 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 A363865 a(n) = Sum_{k = 0..n} binomial(3*n,n-k)^2*binomial(3*n+k,k)^2.
%F A363865 a(n) = Sum_{k = 0..n} binomial(3*n+k,2*k)*binomial(2*k,k)^2*binomial(n+k,2*k).
%F A363865 a(n) = hypergeom([3*n+1, -3*n, n+1, -n], [1, 1, 1], 1)
%F A363865 a(n) = [x^n] 1/(1 - x)*( Legendre_P(3*n,(1 + x)/(1 - x)) )^2 =  [x^(3*n)] 1/(1 - x)*( Legendre_P(n,(1 + x)/(1 - x)) )^2.
%p A363865 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 A363865 seq(A143007(3*n, n), n = 0..20);
%p A363865 # alternative program
%p A363865 seq(simplify(hypergeom([3*n+1, -3*n, n+1, -n], [1, 1, 1], 1)), n = 0..20);
%Y A363865 Cf. A005259, A143007, A363864 - A363871.
%K A363865 nonn,easy
%O A363865 0,2
%A A363865 _Peter Bala_, Jun 25 2023