The following identity can be easily verified using Maple's SumTools:-Summation procedure: for n >= 1,
A005810(n) = binomial(4*n,n) = Sum_{k = 0..3*n} n/(n + k)*binomial(n + k,k).
The binomial coefficients
A005810(n) are known to satisfy the supercongruences
A005810(n*p^r) ==
A005810(n*p^(r-1)) (mod p^(3*r)) for primes p >= 5 and positive integers n and r (see Meštrović, equation 39).
Calculation suggests that the present sequence satisfies the same congruences.
Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for primes p >= 5 and positive integers n and r.
More generally, for m a positive integer, define a sequence u_m by setting u_m(n) = Sum_{k = 0..m*n} n/(n + 2*k)*binomial(n + 2*k,k) for n >= 1.
Then we conjecture that each sequence u_m satisfies the above supercongruences. This is the case m = 3. See
A333093 (case m = 1) and
A352275 (m = 2).
