A365029 - OEIS

%I #10 Oct 06 2023 10:30:51

%S 1,0,28,1035,44876,2104500,104056597,5342503859,282118965580,

%T 15225746918238,836111285393528,46569126655126867,2624469492691484309,

%U 149381829558924820091,8575171411278263451149,495882491862054255448035,28860386333798348100899148,1689200944709783371200111774

%N a(n) = Sum_{k = 0..n} binomial(n+k-1, k)^2 * binomial(2*k-1, n).

%C Conjecture: for all primes p >= 5 the following pair of supercongruences hold:

%C 1) a(p - 1) == a(0) (mod p^3),

%C 2) a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) for all positive integers n and r.

%C More generally, for positive integers A and B with A >= 2, the same supercongruences may hold for the sequence whose n-th term is given by Sum_{k = 0..n} binomial(n+k-1, k)^A * binomial(2*k-1, n)^B.

%F a(n) ~ 2^(6*n+1) / (15*Pi^(3/2)*n^(3/2)). - _Vaclav Kotesovec_, Aug 28 2023

%p seq( add( binomial(n+k-1, k)^2 * binomial(2*k-1, n), k = 0..n), n = 0..20);

%t Table[Sum[Binomial[n+k-1, k]^2 * Binomial[2*k-1, n], {k,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Aug 28 2023 *)

%K nonn,easy

%O 0,3

%A _Peter Bala_, Aug 27 2023