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a(n) = A108625(n,2*n).
7

%I #23 Apr 26 2024 12:09:18

%S 1,5,61,923,15421,272755,5006275,94307855,1811113021,35301145037,

%T 696227550811,13863654392945,278264498108611,5622746346645953,

%U 114268249446672151,2333733620675302423,47868774493665731645,985608360056821004233,20362035153323824192645

%N a(n) = A108625(n,2*n).

%C a(n) = B(n,2*n,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), A363868 (r = 3, s = 1), A363869 (r = 3, s = 2), A363870 (r = 1, s = 3) and A363871 (r = 2, s = 3).

%H G. C. Greubel, <a href="/A363867/b363867.txt">Table of n, a(n) for n = 0..500</a>

%F a(n) = Sum_{k = 0..n} binomial(n, k)^2 * binomial(2*n+k, n).

%F a(n) = Sum_{k = 0..n} (-1)^(n+k)* binomial(n, k)*binomial(2*n+k, n)^2.

%F a(n) = hypergeom( [-n, -2*n, n+1], [1, 1], 1).

%F a(n) = [x^(2*n)] 1/(1 - x)*Legendre_P(n, (1 + x)/(1 - x)).

%F P-recursive: 4*(2*n - 1)^2*n^2*(85*n^2 - 235*n + 163)*a(n) = (29665*n^6 - 141345*n^5 + 264772*n^4 - 249181*n^3 + 124975*n^2 - 31902*n + 3276)*a(n-1) + 4*(2*n - 3)^2*(n-1)^2*(85*n^2 - 65*n + 13)*a(n-2) with a(0) = 1 and a(1) = 5.

%F a(n) = Sum_{k = 0..n} binomial(n, k)*binomial(n+k, k)*binomial(2*n, k). - _Peter Bala_, Feb 25 2024

%F a(n) ~ sqrt(13 + 53/sqrt(17)) * (349 + 85*sqrt(17))^n / (Pi * n * 2^(5*n + 5/2)). - _Vaclav Kotesovec_, Apr 26 2024

%p A108625 := (n, k) -> hypergeom([-n, -k, n+1], [1, 1], 1):

%p seq(simplify(A108625(n, 2*n)), n = 0..18);

%t Table[HypergeometricPFQ[{-n,-2*n,n+1}, {1,1}, 1], {n,0,30}] (* _G. C. Greubel_, Oct 05 2023 *)

%o (Magma)

%o A363867:= func< n | (&+[Binomial(n,j)^2*Binomial(2*n+j,n): j in [0..n]]) >;

%o [A363867(n): n in [0..30]]; // _G. C. Greubel_, Oct 05 2023

%o (SageMath)

%o def A363867(n): return sum(binomial(n,j)^2*binomial(2*n+j,n) for j in range(n+1))

%o [A363867(n) for n in range(31)] # _G. C. Greubel_, Oct 05 2023

%Y Cf. A005258, A099601, A108625, A363864 - A363871.

%K nonn,easy

%O 0,2

%A _Peter Bala_, Jun 27 2023