A179536 a(n) = Sum_{k=0..n} binomial(n,k)^2*binomial(n-k,k)^2*(-324)^k.

1, 1, -1295, -11663, 3732481, 94348801, -12754875599, -662010720335, 43350090126337, 4277886247480321, -117993200918257295, -25968226221675142415, 13219198014412583425, 148460113964113254411265

Offset: 0

Author

Zhi-Wei Sun, Jul 18 2010

Keywords

• sign

Comments

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.

Examples

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

Links

• Zhi-Wei Sun, Open Conjectures on Congruences, preprint, arXiv:0911.5665 [math.NT], 2009-2011.
• Zhi-Wei Sun, On Apery numbers and generalized central trinomial coefficients, preprint, arXiv:1006.2776 [math.NT], 2010-2011.

Programs

Mathematica

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