OFFSET
0,2
COMMENTS
Compare with the Domb numbers A002895, which are defined by A002895(n) = Sum_{k = 0..n} binomial(n,k)^2 * binomial(2*n-2*k,n-k) * binomial(2*k,k).
The supercongruences A002895(n*p^r) == A002895(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and positive integers n and r (see Osburn and Sahu).
We conjecture that the present sequence satisfies the same supercongruences.
More generally, let A >= 2, B and C be positive integers. Then we conjecture that the sequence whose terms are given by Sum_{k = 0..n} binomial(n+k-1,k)^A * binomial(2*n-2*k,n-k)^B * binomial(2*k,k)^C also satisfies the same supercongruences.
LINKS
Peter Bala, A recurrence for A364111.
Robert Osburn and Brundaban Sahu, A supercongruence for generalized Domb numbers, arXiv:1201.6195v2 [math.NT], Functiones et Approximatio. Comment. Math, Vol. 48, No 1, March 2013, 29-36.
FORMULA
a(n) = Sum_{k = 0..n} binomial(-n,k)^2 * binomial(2*n-2*k,n-k) * binomial(2*k,k).
a(n) = binomial(2*n,n)*hypergeom([-n, n, n, 1/2], [1, 1, 1/2 - n], 1).
a(n) ~ 2^(6*n-1) / (sqrt(3) * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Jul 09 2023
MAPLE
seq(add(binomial(n+k-1, k)^2 * binomial(2*n-2*k, n-k) * binomial(2*k, k)), n = 0..20);
# faster program for large n
seq(simplify(binomial(2*n, n)*hypergeom([-n, n, n, 1/2], [1, 1, 1/2 - n], 1)), n = 0..20);
MATHEMATICA
Table[Binomial[2*n, n] * HypergeometricPFQ[{-n, n, n, 1/2}, {1, 1, 1/2 - n}, 1], {n, 0, 20}] (* Vaclav Kotesovec, Jul 09 2023 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Jul 07 2023
STATUS
approved