A179536 - OEIS

%I #10 Nov 09 2019 01:11:55

%S 1,1,-1295,-11663,3732481,94348801,-12754875599,-662010720335,

%T 43350090126337,4277886247480321,-117993200918257295,

%U -25968226221675142415,13219198014412583425,148460113964113254411265

%N a(n) = Sum_{k=0..n} binomial(n,k)^2*binomial(n-k,k)^2*(-324)^k.

%C On Jul 17 2010, _Zhi-Wei Sun_ introduced this sequence and made the following conjecture: If p is an odd prime with (p/13) = (-1/p) = 1 and p = x^2 + 13y^2 with x,y integers, then Sum_{k=0..p-1} a(k) == 4x^2 - 2p (mod p^2); if p is an odd prime with (p/13) = (-1/p) = -1 and 2p = x^2 + 13y^2 with x,y integers, then Sum_{k=0..p-1} a(k) == 2x^2 - 2p (mod p^2); if p > 3 is a prime with (p/13) = -(-1/p), then Sum_{k=0..p-1} a(k) == 0 (mod p^2). He also conjectured that Sum_{k=0..n-1} (260k+237)*a(k) == 0 (mod n) for all n=1,2,3,... and that Sum_{k=0..p-1} (260k+237)*a(k) == p(130(-1/p)+107) (mod p^2) for any prime p > 3.

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/0911.5665">Open Conjectures on Congruences</a>, preprint, arXiv:0911.5665 [math.NT], 2009-2011.

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1006.2776">On Apery numbers and generalized central trinomial coefficients</a>, preprint, arXiv:1006.2776 [math.NT], 2010-2011.

%e For n=2 we have a(2) = 1 + 2^2*(-324) = -1295.

%t a[n_]:=Sum[Binomial[n,k]^2Binomial[n-k,k]^2*(-324)^k,{k,0,n}] Table[a[n],{n,0,25}]

%K sign

%O 0,3

%A _Zhi-Wei Sun_, Jul 18 2010