A357960 a(n) = A005259(n-1)^5 * A005258(n)^6.

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.

Links

Table of n, a(n) for n=1..7.

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);

Crossrefs

Cf. A005258, A005259, A212334, A352655, A357567, A357956, A357957, A357958, A357959.

Sequence in context: A013852 A277185 A343693 * A352034 A321810 A252526

Adjacent sequences: A357957 A357958 A357959 * A357961 A357962 A357963

Keyword

nonn,easy

Author

Peter Bala, Oct 25 2022

Status

approved