A179537 a(n) = Sum_{k=0..n} binomial(n,k)^2*binomial(n-k,k)^2*(-16)^k.

1, 1, -63, -575, 6913, 224001, 420801, -69020223, -918270975, 14596918273, 511845045697, 336721812417, -198449271643391, -2498857696947455, 51614254703660481, 1666776235855331265, -1588877076116525055

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

Table of n, a(n) for n=0..16.

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

Cf. A179536, A179535, A179524, A178790, A178791, A178808, A173774.

Sequence in context: A038858 A307527 A217264 * A184448 A228262 A340902

Adjacent sequences: A179534 A179535 A179536 * A179538 A179539 A179540

Keyword

sign

Author

Zhi-Wei Sun, Jul 18 2010

Status

approved