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A179524
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a(n) = Sum_{k=0..n} (-4)^k*binomial(n,k)^2*binomial(n-k,k)^2.
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2
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1, 1, -15, -143, 1, 12801, 100401, -555855, -16006143, -69903359, 1371541105, 20881151985, 5878439425, -2725373454335, -25310084063055, 145439041081137, 4851621446905857, 23952290336559105, -470461357757965071, -7793050905481342863, -4149447893184517119
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OFFSET
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0,3
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COMMENTS
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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).
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LINKS
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FORMULA
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a(n) = Sum_{k=0..[n/2]} (-4)^k*binomial(n,2k)^2*binomial(2k,k)^2.
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EXAMPLE
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For n=3 we have a(3)=1-4*3^2*2^2=-143.
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MATHEMATICA
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W[n_]:=Sum[(-4)^k*Binomial[n, k]^2*Binomial[n-k, k]^2, {k, 0, n}] Table[W[n], {n, 0, 50}]
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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