A364111 a(n) = Sum_{k = 0..n} binomial(n+k-1,k)^2 * binomial(2*n-2*k,n-k) * binomial(2*k,k).

1, 4, 76, 2560, 106060, 4864504, 237354880, 12079462560, 633885607500, 34050190896040, 1863047125801576, 103465470769890112, 5817117095161011328, 330450303019252600240, 18937657945720403830240, 1093557503049551583194560, 63566414131528881235953228, 3716526456851323626808570632

Offset: 0

Author

Peter Bala, Jul 07 2023

Keywords

• nonn
• easy

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.

Crossrefs

Cf. A002895, A362676.

Programs

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 *)

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