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%I #24 Jan 24 2019 08:14:47
%S 1,-7,117,-2441,57449,-1453635,38609845,-1061792695,29973352185,
%T -863536596143,25288254409373,-750531594051981,22525211241191881,
%U -682459907754004723,20845409947239778533,-641211780685502724425
%N The arithmetic mean of (2k+1)(-1)^k*A_k (k=0,...,n-1), where A_0, A_1,... are Apéry numbers given by A005259.
%C On Jun 14 2010, _Zhi-Wei Sun_ conjectured that the number a(n) = n^{-1}*Sum_{k=0..n-1} (2*k+1)*(-1)^k*A_k is always an integer and that a(p) = p(p/3) (mod p^3) for any prime p>3. He also formulated the following conjecture: If p=1 (mod 3) is a prime and p = x^2 + 3y^2 with x,y integers then Sum_{k=0..p-1} (-1)^k*A_k = 4*x^2 - 2*p (mod p^2); if p=2 (mod 3) is a prime then Sum_{k=0..p-1 }(-1)^k*A_k=0 (mod p^2).
%H G. C. Greubel, <a href="/A178791/b178791.txt">Table of n, a(n) for n = 1..500</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1006.2776">Arithmetic properties of Apery numbers and central Delannoy numbers</a>, arXiv:1006.2776 [math.NT], 2010-2011.
%F G.f.: apart from the minus signs (just replace x by -x) the generating function is 2*x*G/(x+1) + Int((x-1)*G/(x+1)^2, x) where G is the generating function of A005259. - _Mark van Hoeij_, May 07 2013
%F a(n) ~ -(-1)^n * 2^(3/4) * (1 + sqrt(2))^(4*n) / (24 * (Pi*n)^(3/2)). - _Vaclav Kotesovec_, Jan 24 2019
%e For n=3 we have a(3) = (A_0 - 3A_1 + 5A_2)/3 = (1 - 3*5 + 5*73)/3 = 117.
%p G := (-1/2)*(3*x-3+(x^2-34*x+1)^(1/2))*(x+1)^(-2)*hypergeom([1/3, 2/3], [1], (-1/2)*(x^2-7*x+1)*(x+1)^(-3)*(x^2-34*x+1)^(1/2)+(1/2)*(x^3+30*x^2 -24*x+1)*(x+1)^(-3))^2;
%p ogf := 2*x*G/(x+1)+Int((x-1)*G/(x+1)^2,x);
%p series(ogf, x=0, 25);
%p series(-subs(x=-x,%), x=0, 25); # _Mark van Hoeij_, May 07 2013
%t Apery[n_]:= Sum[Binomial[n+k,k]^2Binomial[n,k]^2,{k,0,n}]; AA[n_]:= Sum[(2k+1)(-1)^k*Apery[k],{k,0,n-1}]/n; Table[AA[n],{n,25}]
%o (PARI) A(n) = sum(k=0, n, (binomial(n, k)*binomial(n+k, k))^2); \\ A005259
%o a(n) = sum(k=0, n-1, (2*k+1)*(-1)^k*A(k))/n; \\ _Michel Marcus_, Jan 24 2019
%Y Cf. A178790, A005259, A173774.
%K sign
%O 1,2
%A _Zhi-Wei Sun_, Jun 14 2010
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