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A357960
a(n) = A005259(n-1)^5 * A005258(n)^6.
5
729, 147018378125, 20917910914764786689697, 24148107115850058575342740485778125, 79477722547796770983047586179643766765851375729, 492664048531500749211923278756418311980637289373757041378125, 4671227340507161302417161873394448514470099313382652883508175438056640625
OFFSET
1,1
COMMENTS
Conjectures:
1) a(p) == a(1) (mod p^5) for all primes p >= 3 (checked up to p = 271).
2) a(p^r) == a(p^(r-1)) ( mod p^(3*r+3) ) for r >= 2 and for all primes p >= 3. These are stronger supercongruences than those satisfied separately by the two types of Apéry numbers A005258 and A005259.
FORMULA
a(n) = ( Sum_{k = 0..n-1} binomial(n-1,k)^2*binomial(n+k-1,k)^2 )^5 * ( Sum_{k = 0..n} binomial(n,k)^2*binomial(n+k,k) )^6.
a(n*p^r) == a(n*p^(r-1)) ( mod p^(3*r) ) for positive integers n and r and for all primes p >= 5.
MAPLE
seq( add(binomial(n-1, k)^2*binomial(n+k-1, k)^2, k = 0..n-1)^5 * add(binomial(n, k)^2*binomial(n+k, k), k = 0..n)^6, n = 1..20);
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Oct 25 2022
STATUS
approved