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A363989
a(n) = Sum_{k = 0..n} (-4)^(n-k)*binomial(n,k)*binomial(4*n+k,k)*binomial(2*k,k).
2
1, 6, 142, 3732, 108750, 3345256, 106702756, 3491407296, 116430027470, 3939976259340, 134891827384392, 4662318475294992, 162421061711817252, 5696013237211704192, 200894768178332121280, 7120345964204221311232, 253453190156679801719502
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.
FORMULA
a(n) = (-4)^n*hypergeom([-n, 4*n+1, 1/2], [1, 1], 1)).
a(n) ~ 2^(10*n - 1/2) / (Pi * n * 3^(3*n)). - Vaclav Kotesovec, Jul 17 2023
MAPLE
seq(add((-4)^(n-k)*binomial(n, k)*binomial(4*n+k, k)*binomial(2*k, k), k = 0..n), n = 0..20);
# alternative faster program for large n
seq(simplify((-4)^n*hypergeom([-n, 4*n+1, 1/2], [1, 1], 1)), n = 0..20);
MATHEMATICA
Table[(-4)^n*HypergeometricPFQ[{-n, 4*n+1, 1/2}, {1, 1}, 1], {n, 0, 20}] (* Vaclav Kotesovec, Jul 17 2023 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Jul 02 2023
STATUS
approved