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A364243 a(n) = A108625(2*n-1, n-1) for n >= 1.

%I A364243 #13 Jul 18 2023 04:09:55
%S A364243 1,13,271,6637,176251,4914427,141573251,4174790893,125288929171,
%T A364243 3811804637263,117248436333601,3638993432201563,113790712076898871,
%U A364243 3580847269415337487,113299135244467189771,3601766951734150461677,114973519461796962202051
%N A364243 a(n) = A108625(2*n-1, n-1) for n >= 1.
%C A364243 The sequence of Apéry numbers A005258 forms the main diagonal of A108625, i.e., A005258(n) = A108625(n, n). The Apéry numbers satisfy the supercongruences A005258(n*p^r) == A005258(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 A364243 More generally, for positive integers r and s, the sequence defined by a(r,s;n) = A108625(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 A363867.
%F A364243 a(n) = Sum_{k = 0..n-1} binomial(2*n-1,k)^2 * binomial(3*n-2-k,2*n-1) for n >= 1.
%F A364243 a(n) = Sum_{k = 0..n-1} (-1)^k * binomial(2*n-1,k) * binomial(3*n-2-k,2*n-1)^2 for n >= 1.
%F A364243 a(n) = binomial(2*n-1,n-1)^2 * hypergeom([1 - n, 1 - n, 2*n], [1 + n, 1 + n], 1).
%F A364243 a(n) = (-1)^(n+1) * binomial(2*n-1,n-1) * hypergeom([1 - n, 2*n, 2*n], [1, 1 + n], 1).
%F A364243 P-recursive: (n-1)^2*(2*n-1)^2*(48*n^2-162*n+137)*a(n) = (6528*n^6-48144*n^5+144800*n^4-226714*n^3+194349*n^2-86299*n+15515)*a(n-1) - (2*n-3)^2*(n-2)^2*(48*n^2-66*n+23)*a(n-2) with a(1) = 1 and a(2) = 13.
%F A364243 a(n) ~ (1 + sqrt(2))^(4*n - 3/2) / (2^(9/4)*Pi*n). - _Vaclav Kotesovec_, Jul 16 2023
%p A364243 seq(add(binomial(2*n-1,k)^2 * binomial(3*n-2-k,2*n-1), k = 0..n-1), n = 1..20);
%p A364243 # alternative program
%p A364243 seq( simplify(binomial(2*n-1,n-1)^2 * hypergeom([1 - n, 1 - n, 2*n], [1 + n, 1 + n], 1)), n = 1..20);
%t A364243 Table[Binomial[2*n-1, n-1]^2 * HypergeometricPFQ[{1 - n, 1 - n, 2*n}, {1 + n, 1 + n}, 1], {n, 1, 20}] (* _Vaclav Kotesovec_, Jul 16 2023 *)
%Y A364243 Cf. A005258, A108625, A363867, A364244.
%K A364243 nonn,easy
%O A364243 1,2
%A A364243 _Peter Bala_, Jul 16 2023