OFFSET
0,2
COMMENTS
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.
More generally, for positive integers r and s the sequence {A108625(r*n, s*n) : n >= 0} satisfies the same supercongruences.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..400
Peter Bala, A recurrence for A363871
FORMULA
a(n) = Sum_{k = 0..2*n} binomial(2*n, k)^2 * binomial(5*n-k, 2*n).
a(n) = Sum_{k = 0..2*n} (-1)^k * binomial(2*n, k)*binomial(5*n-k, 2*n)^2.
a(n) = hypergeometric3F2([-2*n, -3*n, 2*n+1], [1, 1], 1).
a(n) = [x^(3*n)] 1/(1 - x)*Legendre_P(2*n, (1 + x)/(1 - x)).
a(n) ~ sqrt(1700 + 530*sqrt(10)) * (98729 + 31220*sqrt(10))^n / (120 * Pi * n * 3^(6*n)). - Vaclav Kotesovec, Feb 17 2024
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
MAPLE
MATHEMATICA
Table[HypergeometricPFQ[{-2*n, -3*n, 2*n+1}, {1, 1}, 1], {n, 0, 30}] (* G. C. Greubel, Oct 05 2023 *)
PROG
(Magma)
A363871:= func< n | (&+[Binomial(2*n, j)^2*Binomial(5*n-j, 2*n): j in [0..2*n]]) >;
[A363871(n): n in [0..30]]; // G. C. Greubel, Oct 05 2023
(SageMath)
def A363871(n): return sum(binomial(2*n, j)^2*binomial(5*n-j, 2*n) for j in range(2*n+1))
[A363871(n) for n in range(31)] # G. C. Greubel, Oct 05 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Jun 27 2023
STATUS
approved