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A364244 a(n) = A143007(2*n-1, n-1) for n >= 1.

%I A364244 #21 Nov 04 2023 02:52:57
%S A364244 1,25,1441,107353,9073501,826861993,79219824685,7865844936025,
%T A364244 802198564524325,83532710607121525,8844234718023010681,
%U A364244 949244022625120188265,103044177225432902852641,11293765432962617876667253,1248038875078327818254657941
%N A364244 a(n) = A143007(2*n-1, n-1) for n >= 1.
%C A364244 The sequence of Apéry numbers A005259 forms the main diagonal of A143007, i.e., A005259(n) = A143007(n, n). The Apéry numbers satisfy the supercongruences A005259(n*p^r) == A005259(n^p^(r-1)) (mod p^(3*r)) for all primes p >= 5 and positive integers n and r. We conjecture that the present sequence satisfies the same supercongruences.
%C A364244 More generally, for positive integers r and s, the sequence defined by a(r,s;n) = A143007(r*n - 1, s*n - 1) may also satisfy the same supercongruences. This is the case r = 2, s = 1. Compare with the comments in A363864.
%F A364244 a(n) = Sum_{k = 0..n-1} binomial(2*n-1, k)^2 * binomial(3*n-2-k, 2*n-1)^2.
%F A364244 a(n) = hypergeom([2*n, 1 - 2*n, n, 1 - n], [1, 1, 1], 1).
%F A364244 P-recursive: 2*(n-1)^3*(2*n-1)^3*(440*n^3-2178*n^2+3600*n-1987)*a(n) = (865920*n^9 - 9481824*n^8 + 45492136*n^7 - 125359294*n^6 + 218361816*n^5 - 249018285*n^4 + 185709390*n^3 - 87271191*n^2 + 23447876*n - 2745998)*a(n-1) -  2*(2*n-3)^3*(n-2)^3*(440*n^3-858*n^2+564*n-125)*a(n-2) with a(1) = 1 and a(2) = 25.
%F A364244 a(n) ~ phi^(10*n - 4) / (2^(5/2) * 5^(1/4) * (Pi*n)^(3/2)), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Jul 16 2023
%p A364244 seq( add(binomial(2*n-1, k)^2 * binomial(3*n-2-k, 2*n-1)^2, k = 0..n-1), n = 1..20);
%p A364244 # alternative program
%p A364244 seq(simplify(hypergeom([2*n, 1 - 2*n, n, 1 - n], [1, 1, 1], 1)), n = 1..20);
%t A364244 Table[HypergeometricPFQ[{2*n, 1 - 2*n, n, 1 - n}, {1, 1, 1}, 1], {n, 1, 20}] (* _Vaclav Kotesovec_, Jul 16 2023 *)
%Y A364244 Cf. A005259, A143007, A363864, A364243.
%K A364244 nonn,easy
%O A364244 1,2
%A A364244 _Peter Bala_, Jul 16 2023
%E A364244 Offset changed by _Georg Fischer_, Nov 03 2023