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%I #21 Jan 30 2020 21:29:17
%S -1,0,3,-5,-3,26,-35,-48,245,-248,-639,2355,-1573,-7890,22555,-6685,
%T -93075,212280,27625,-1061415,1938855,1276550,-11763465,16906450,
%U 23324507,-126971664,136840575,343314517,-1334857995,965192298
%N G.f. (1-x)/(2*sqrt(5*x^2 + 2*x + 1)) - 1/2.
%F a(n) = Sum_{k=ceiling(n/2)..n} (-1)^k*binomial(k,n-k)*binomial(n-1,n-k).
%F a(n) = (-1)^k*binomial(n-1, n-k)*binomial(k, n-k)*hypergeom([1, 1+k, -n+k, -n+k], [k, 1/2-n/2+k, 1-n/2+k], -1/4) where k = ceiling(n/2). - _Peter Luschny_, Nov 09 2014
%F D-finite with recurrence: n*a(n) + (n+1)*a(n-1) + 3*n*a(n-2) + 5*(-n+3)*a(n-3) = 0. - _R. J. Mathar_, May 22 2019
%F n*(4*n-7)*a(n) + 2*(4*n-1)*(n-2)*a(n-1) + 5*(4*n-3)*(n-2)*a(n-2) = 0. - _R. J. Mathar_, May 22 2019
%p T := (n, k) -> (-1)^k*binomial(n-1, n-k)*binomial(k, n-k)*
%p hypergeom([1, 1+k, -n+k, -n+k], [k, 1/2-n/2+k, 1-n/2+k], -1/4):
%p seq(simplify(T(n, ceil(n/2))), n=1..30); # _Peter Luschny_, Nov 09 2014
%t Rest[CoefficientList[Series[(1 - x) / (2 Sqrt[5 x^2 + 2 x + 1]) - 1/2, {x, 0, 40}], x]] (* _Vincenzo Librandi_, Nov 08 2014 *)
%o (Maxima) a(n):=sum((-1)^k*binomial(k,n-k)*binomial(n-1,n-k),k,ceiling(n/2),n);
%o (PARI) a(n) = sum(k=ceil(n/2), n, (-1)^k*binomial(k,n-k)*binomial(n-1,n-k)); \\ _Michel Marcus_, Nov 09 2014
%K sign
%O 1,3
%A _Vladimir Kruchinin_, Nov 08 2014
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