OFFSET
0,2
COMMENTS
Define S_m(n) = the numerator of Sum_{k = 1..n} (-1)^(n+k)*(1/k^m)*binomial(n,k)* binomial(n+k,k)^2, so that S_0(n) = -1 + A005258(n), one of the two types of Apéry numbers. The present sequence is the case m = 1. See A357561 for the case m = 3.
Conjectures:
1) for even m >= 2, S_m(p-1) == 0 (mod p^3) for all primes p > m + 3.
2) for odd m >= 1, S_m(p-1) == 0 (mod p^4) for all primes p > m + 4.
LINKS
A. Straub, Multivariate Apéry numbers and supercongruences of rational functions, arXiv:1401.0854 [math.NT] (2014).
FORMULA
Conjecture: a(p-1) == 0 (mod p^4) for all primes p >= 7 (checked up to p = 499).
Note: the Apéry numbers B(n) = A005258(n) = Sum_{k = 0..n} (-1)^(n+k)* binomial(n,k)*binomial(n+k,k)^2 satisfy the supercongruences B(p-1) == 1 (mod p^3) for all primes p >= 5 (see, for example, Straub, Example 3.4).
EXAMPLE
Example of a supercongruence:
p = 19: a(19 - 1) = 51051733540797155872 = (2^5)*(19^4)*12241823444801 == 0 (mod 19^4).
MAPLE
seq( numer(add( (-1)^(n+k) * (1/k) * binomial(n, k) * binomial(n+k, k)^2, k = 1..n )), n = 0..20 );
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Oct 04 2022
STATUS
approved