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%I #11 Nov 20 2015 16:34:16
%S 1,-1,1,-11,65,-223193,52003,-4741887,4535189795,-25273021529,
%T 1847538284735,-13861021151998879,70722327129418887,
%U -268605133000589500717,23799858495522904017785,-108128513649040594935009169,1403426321169925536031927183,-27867021017469762051006316943497
%N Numerators of coefficients c(n) in asymptotic expansion of Sum_{m=1..k} sqrt(m) ~ zeta(-1/2) + (2/3)*k^(3/2) + (-1/2)*k^(1/2) + Sum_{n=0..inf} c(n)/k^(2*n+1/2).
%e Sum_{m=1..k} sqrt(m) = zeta(-1/2) + (2/3)*k^(3/2) + (-1/2)*k^(1/2) + (1/24)/k^(1/2) + (-1/1920)/k^(5/2) + (1/9216)/k^(9/2) + (-11/163840)/k^(13/2) + (65/786432)/k^(17/2) + O(1/k^(21/2)).
%t nmax = 20; Table[Numerator[Simplify[Normal[Series[HarmonicNumber[k, -1/2] - (2*k^(3/2)/3 + Sqrt[k]/2 + Zeta[-1/2]), {k, Infinity, 2*nmax}]][[j]]*k^((4*j - 3)/2), k > 0]], {j, 1, nmax}] (* _Vaclav Kotesovec_, Nov 20 2015 *)
%K sign
%O 0,4
%A _Vladimir Reshetnikov_, Nov 09 2015
%E More terms from _Vaclav Kotesovec_, Nov 20 2015
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