OFFSET
0,3
COMMENTS
Conjecture 1: the supercongruence a(p) == a(1) (mod p^5) holds for all primes p >= 7 (checked up to p = 199).
Conjecture 2: for r >= 2, the supercongruence a(p^r) == a(p^(r-1)) (mod p^(4*r+1)) holds for all primes p >= 7.
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..650
Peter Bala, Recurrence equation for A361713
FORMULA
a(n) ~ C*(12*sqrt(2) + 17)^n/n^(3/2), where C = 1/(2^(5/4)*Pi^(3/2)).
a(n) = hypergeom([-n, -n, n, n], [1, 1, 1], 1) - binomial(2*n-1, n)^2. This is another way to write the first formula. - Peter Luschny, Mar 27 2023
MAPLE
seq(add(binomial(n, k)^2*binomial(n+k-1, k)^2, k = 0..n-1), n = 0..25);
# Alternative:
A361713 := n -> hypergeom([-n, -n, n, n], [1, 1, 1], 1) - binomial(2*n - 1, n)^2:
seq(simplify(A361713(n)), n = 0..18); # Peter Luschny, Mar 27 2023
MATHEMATICA
A361713[n_] := HypergeometricPFQ[{-n, -n, n, n}, {1, 1, 1}, 1] - Binomial[2*n-1, n]^2; Array[A361713, 20, 0] (* Paolo Xausa, Jul 11 2024 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Mar 21 2023
STATUS
approved