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A179537 a(n) = Sum_{k=0..n} binomial(n,k)^2*binomial(n-k,k)^2*(-16)^k. 0
1, 1, -63, -575, 6913, 224001, 420801, -69020223, -918270975, 14596918273, 511845045697, 336721812417, -198449271643391, -2498857696947455, 51614254703660481, 1666776235855331265, -1588877076116525055 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
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.
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.
EXAMPLE
For n=2 we have a(2)=1+2^2*(-16)=-63.
MATHEMATICA
a[n_]:=Sum[Binomial[n, k]^2Binomial[n-k, k]^2*(-16)^k, {k, 0, n}] Table[a[n], {n, 0, 25}]
CROSSREFS
Sequence in context: A038858 A307527 A217264 * A184448 A228262 A340902
KEYWORD
sign
AUTHOR
Zhi-Wei Sun, Jul 18 2010
STATUS
approved

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Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)