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A179537
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a(n) = Sum_{k=0..n} binomial(n,k)^2*binomial(n-k,k)^2*(-16)^k.
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0
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1, 1, -63, -575, 6913, 224001, 420801, -69020223, -918270975, 14596918273, 511845045697, 336721812417, -198449271643391, -2498857696947455, 51614254703660481, 1666776235855331265, -1588877076116525055
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OFFSET
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0,3
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COMMENTS
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On July 17, 2010 Zhi-Wei Sun introduced this sequence and made the following conjecture: If p is a prime with (p/7)=1 and p=x^2+7y^2 with x,y integers, then sum_{k=0}^{p-1}(-1)^k*a(k)=4x^2-2p (mod p^2); if p is a prime with (p/7)=-1, then sum_{k=0}^{p-1}(-1)^k*a(k)=0 (mod p^2). He also conjectured that sum_{k=0}^{n-1}(42k+37)(-1)^k*a(k)=0 (mod n) for all n=1,2,3,... and that sum_{k=0}^{p-1}(42k+37)(-1)^k*a(k)=p(21(p/7)+16) (mod p^2) for any prime p.
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LINKS
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EXAMPLE
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For n=2 we have a(2)=1+2^2*(-16)=-63.
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MATHEMATICA
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a[n_]:=Sum[Binomial[n, k]^2Binomial[n-k, k]^2*(-16)^k, {k, 0, n}] Table[a[n], {n, 0, 25}]
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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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