OFFSET
0,2
COMMENTS
Cf. A005258(n) = Sum_{k = 0..n} (-1)^k*binomial(n, k)*binomial(n+k, k)^2.
The sequence of Apéry numbers A005258 satisfies the pair of supercongruences
1) A005258(n*p^r) == A005258(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 5 and all positive integers n and r
and
2) A005258(n*p^r - 1) == A005258(n*p^(r-1) - 1) (mod p^(3*r)) for all primes p >= 5 and all positive integers n and r.
We conjecture that the present sequence satisfies the same pair of supercongruences. Some examples are given below.
FORMULA
Examples of supercongruences:
a(11) - a(1) = 306174745758226208537 - 3 = 2*(11^3)*17*79367*85245689663 == 0 (mod 11^3).
a(10) - a(0) = 2329136571942011877 - 1 = (2^2)*(11^3)*17011*25717400209 == 0 (mod 11^3).
MAPLE
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Sep 25 2024
STATUS
approved