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%I #7 Jan 12 2019 08:06:24
%S 1,1,-15,-143,1,12801,100401,-555855,-16006143,-69903359,1371541105,
%T 20881151985,5878439425,-2725373454335,-25310084063055,
%U 145439041081137,4851621446905857,23952290336559105,-470461357757965071,-7793050905481342863,-4149447893184517119
%N a(n) = Sum_{k=0..n} (-4)^k*binomial(n,k)^2*binomial(n-k,k)^2.
%C 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).
%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.
%F a(n) = Sum_{k=0..[n/2]} (-4)^k*binomial(n,2k)^2*binomial(2k,k)^2.
%e For n=3 we have a(3)=1-4*3^2*2^2=-143.
%t W[n_]:=Sum[(-4)^k*Binomial[n,k]^2*Binomial[n-k,k]^2,{k,0,n}] Table[W[n],{n,0,50}]
%Y Cf. A005259, A178790, A178791, A178808, A179508, A173774.
%K sign
%O 0,3
%A _Zhi-Wei Sun_, Jul 17 2010
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