A179524 a(n) = Sum_{k=0..n} (-4)^k*binomial(n,k)^2*binomial(n-k,k)^2.

1, 1, -15, -143, 1, 12801, 100401, -555855, -16006143, -69903359, 1371541105, 20881151985, 5878439425, -2725373454335, -25310084063055, 145439041081137, 4851621446905857, 23952290336559105, -470461357757965071, -7793050905481342863, -4149447893184517119

Offset: 0

Author

Zhi-Wei Sun, Jul 17 2010

Keywords

• sign

Comments

On July 1, 2010 Zhi-Wei Sun introduced this sequence and made the following conjecture: If p is a prime with p=1,9 (mod 20) and p=x^2+5y^2 with x,y integers, then sum_{k=0}^{p-1}a(k)=4x^2-2p (mod p^2); if p is a prime with p=3,7 (mod 20) and 2p=x^2+5y^2 with x,y integers, then sum_{k=0}^{p-1}a(k)=2x^2-2p (mod p^2); if p is a prime with p=11,13,17,19 (mod 20), then sum_{k=0}^{p-1}w_k=0 (mod p^2). He also conjectured that sum_{k=0}^{n-1}(20k+17)w_k=0 (mod n) for all n=1,2,3,... and that sum_{k=0}^{p-1}(20k+17)w_k=p(10(-1/p)+7) (mod p^2) for any odd prime p. Sun also formulated similar conjectures for some sequences similar to a(n).

Examples

			For n=3 we have a(3)=1-4*3^2*2^2=-143.
		

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.

Crossrefs

Cf. A005259, A178790, A178791, A178808, A179508, A173774.

Programs

Mathematica

W[n_]:=Sum[(-4)^k*Binomial[n,k]^2*Binomial[n-k,k]^2,{k,0,n}] Table[W[n],{n,0,50}]

Formula

a(n) = Sum_{k=0..[n/2]} (-4)^k*binomial(n,2k)^2*binomial(2k,k)^2.