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%I #17 Feb 26 2024 09:14:11
%S 1,37,5321,980407,201186025,43859522037,9939874413899,
%T 2314357836947571,549694303511409641,132569070434503802605,
%U 32360243622138480889321,7977001183875449759759807,1982402220908671654519130731,496031095735572731850517509727
%N a(n) = A108625(2*n, 3*n).
%C a(n) = B(2*n, 3*n, 2*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 More generally, for positive integers r and s the sequence {A108625(r*n, s*n) : n >= 0} satisfies the same supercongruences.
%C For other cases, see A099601 (r = 2, s = 1), A363867 (r = 1, s = 2), A363868 (r = 3, s = 1), A363869 (r = 3, s = 2) and A363870 (r = 1, s = 3).
%H G. C. Greubel, <a href="/A363871/b363871.txt">Table of n, a(n) for n = 0..400</a>
%H Peter Bala, <a href="/A363871/a363871.pdf">A recurrence for A363871</a>
%F a(n) = Sum_{k = 0..2*n} binomial(2*n, k)^2 * binomial(5*n-k, 2*n).
%F a(n) = Sum_{k = 0..2*n} (-1)^k * binomial(2*n, k)*binomial(5*n-k, 2*n)^2.
%F a(n) = hypergeometric3F2([-2*n, -3*n, 2*n+1], [1, 1], 1).
%F a(n) = [x^(3*n)] 1/(1 - x)*Legendre_P(2*n, (1 + x)/(1 - x)).
%F a(n) ~ sqrt(1700 + 530*sqrt(10)) * (98729 + 31220*sqrt(10))^n / (120 * Pi * n * 3^(6*n)). - _Vaclav Kotesovec_, Feb 17 2024
%F a(n) = Sum_{k = 0..2*n} binomial(2*n, k) * binomial(3*n, k) * binomial(2*n+k, k). - _Peter Bala_, Feb 26 2024
%p A108625 := (n, k) -> hypergeom([-n, -k, n+1], [1, 1], 1):
%p seq(simplify(A108625(2*n, 3*n)), n = 0..20);
%t Table[HypergeometricPFQ[{-2*n,-3*n,2*n+1}, {1,1}, 1], {n,0,30}] (* _G. C. Greubel_, Oct 05 2023 *)
%o (Magma)
%o A363871:= func< n | (&+[Binomial(2*n,j)^2*Binomial(5*n-j,2*n): j in [0..2*n]]) >;
%o [A363871(n): n in [0..30]]; // _G. C. Greubel_, Oct 05 2023
%o (SageMath)
%o def A363871(n): return sum(binomial(2*n,j)^2*binomial(5*n-j,2*n) for j in range(2*n+1))
%o [A363871(n) for n in range(31)] # _G. C. Greubel_, Oct 05 2023
%Y Cf. A005258, A099601, A108625, A363864 - A363870.
%K nonn,easy
%O 0,2
%A _Peter Bala_, Jun 27 2023
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