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A363869 a(n) = A108625(3*n, 2*n).

%I A363869 #18 Apr 27 2024 07:58:19
%S A363869 1,55,12559,3685123,1205189519,418856591055,151353475289275,
%T A363869 56193989426243199,21283943385478109071,8185785098679048061837,
%U A363869 3186604888590691870779559,1252744279186835597251089055,496508748101370063304243706939,198134918989716743103591120933103
%N A363869 a(n) = A108625(3*n, 2*n).
%C A363869 a(n) = B(3*n, 2*n, 3*n) in the notation of Straub, equation 24. It follows from Straub, Theorem 3.2, that the supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(3*k)) hold for all primes p >= 5 and all positive integers n and k.
%C A363869 More generally, for positive integers r and s the sequence {A108625(r*n, s*n) : n >= 0} satisfies the same supercongruences.
%C A363869 For other cases, see A099601 (r = 2, s = 1), A363867 (r = 1, s = 2), A363868 (r = 3, s = 1), A363870 (r = 1, s = 3) and A363871 (r = 2, s = 3).
%H A363869 Paolo Xausa, <a href="/A363869/b363869.txt">Table of n, a(n) for n = 0..350</a>
%H A363869 Peter Bala, <a href="/A363869/a363869.pdf">A recurrence for A363869</a>
%F A363869 a(n) = Sum_{k = 0..2*n} binomial(3*n, k)^2 * binomial(5*n-k, 3*n).
%F A363869 a(n) = Sum_{k = 0..2*n} (-1)^k * binomial(3*n, k)*binomial(5*n-k, 3*n)^2.
%F A363869 a(n) = hypergeom( [-2*n, -3*n, 3*n+1], [1, 1], 1).
%F A363869 a(n) = [x^(2*n)] 1/(1 - x)*Legendre_P(3*n, (1 + x)/(1 - x)).
%F A363869 a(n) ~ 2^(4*n) * 3^(3*n) / (sqrt(5)*Pi*n). - _Vaclav Kotesovec_, Apr 27 2024
%p A363869 A108625 := (n, k) -> hypergeom([-n, -k, n+1], [1, 1], 1):
%p A363869 seq(simplify(A108625(3*n, 2*n)), n = 0..20);
%t A363869 A363869[n_] := HypergeometricPFQ[{-2*n, -3*n, 3*n + 1}, {1, 1}, 1];
%t A363869 Array[A363869, 20, 0] (* _Paolo Xausa_, Feb 26 2024 *)
%Y A363869 Cf. A005258, A099601, A108625, A363864 - A363871.
%K A363869 nonn,easy
%O A363869 0,2
%A A363869 _Peter Bala_, Jun 27 2023