A363988 a(n) = Sum_{k = 0..n} (-4)^(n-k)*binomial(n,k)*binomial(3*n+2*k,2*k)* binomial(2*k,k).

1, 16, 828, 53836, 3879404, 296396016, 23517939996, 1916072466688, 159188357217516, 13425731614346272, 1145885291754711328, 98752981735587825288, 8579149369628730276860, 750365776730717473307920, 66009615160057048401092544, 5835864811138398925049262336

Offset

0,2

Comments

The sequence of Franel numbers A000172 satisfies the identity A000172(n) = Sum_{k = 0..n} (-4)^(n-k)*binomial(n,k)*binomial(n+2*k,2*k)*binomial(2*k,k). The present sequence comes from a modification of the right-hand side of the identity.

The Franel numbers satisfy the supercongruences u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 5 and positive integers n and r. We conjecture that the present sequence satisfies the same supercongruences.

Links

Table of n, a(n) for n=0..15.

Formula

a(n) = (-4)^n*hypergeom([-n, (3*n+1)/2, (3*n+2)/2], [1, 1], 1)).

a(n) ~ sqrt(5 + 9*sqrt(3/11)) * 3^(n - 1/2) * (63 + 11*sqrt(33))^n / (Pi * n * 2^(2*n + 3/2)). - Vaclav Kotesovec, Jul 17 2023

Maple

seq(add((-4)^(n-k)*binomial(n, k)*binomial(3*n+2*k, 2*k)*binomial(2*k, k), k = 0..n), n = 0..20);
# alternative faster program for large n
seq(simplify((-4)^n*hypergeom([-n, (3*n+1)/2, (3*n+2)/2], [1, 1], 1)), n = 0..20);

Mathematica

Table[(-4)^n*HypergeometricPFQ[{-n, (3*n+1)/2, (3*n+2)/2}, {1, 1}, 1], {n, 0, 20}] (* Vaclav Kotesovec, Jul 17 2023 *)

Crossrefs

Cf. A000172, A362676, A363985 - A363990.

Sequence in context: A159872 A220750 A166187 * A223113 A145243 A264125

Adjacent sequences: A363985 A363986 A363987 * A363989 A363990 A363991

Keyword

nonn,easy

Author

Peter Bala, Jul 02 2023

Status

approved