A357956 - OEIS

%I #13 Nov 03 2022 04:51:27

%S 3,19,327,6931,162503,4072519,107094207,2919528211,81819974343,

%T 2343260407519,68285241342827,2018360803903111,60366625228511423,

%U 1823565812734012639,55557838850469305327,1705172303553678726931,52672608711829111519943,1636296668756812403477839,51088496012515356589705107

%N a(n) = 5*A005259(n) - 2*A005258(n).

%C Conjectures:

%C 1) a(p - 1) == 3 (mod p^5) for all primes p >= 5.

%C 2) a(p^r - 1) == a(p^(r-1) - 1) ( mod p^(3*r+3) ) for r >= 2 and for all primes p >= 5.

%C These are stronger supercongruences than those satisfied separately by the two types of Apéry numbers A005258 and A005259. Cf. A357567.

%H Peter Bala, <a href="/A357956/a357956.pdf">Some conjectural supercongruences for the Apéry numbers.</a>

%F a(n) = 5*Sum_{k = 0..n} binomial(n,k)^2*binomial(n+k,k)^2 - 2*Sum_{k = 0..n} binomial(n,k)^2*binomial(n+k,k).

%F a(n*p^r - 1) == a(n*p^(r-1) - 1) ( mod p^(3*r) ) for positive integers n and r and for all primes p >= 5.

%F a(n) = 5*hypergeom([-n, -n, 1 + n, 1 + n], [1, 1, 1], 1) - 2*hypergeom([1 + n, -n, -n], [1, 1], 1). - _Peter Luschny_, Nov 01 2022

%p seq(add(5*binomial(n,k)^2*binomial(n+k,k)^2 - 2*binomial(n,k)^2* binomial(n+k,k), k = 0..n), n = 0..20);

%p # Alternatively:

%p a := n -> 5*hypergeom([-n, -n, 1 + n, 1 + n], [1, 1, 1], 1) - 2*hypergeom([1 + n, -n, -n], [1, 1], 1): seq(simplify(a(n)), n = 0..18); # _Peter Luschny_, Nov 01 2022

%Y Cf. A005258, A005259, A212334, A352655, A357567, A357568, A357569, A357957, A357958, A357959, A357960.

%K nonn,easy

%O 0,1

%A _Peter Bala_, Oct 24 2022