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A260151
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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).
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0
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1, -1, 1, -11, 65, -223193, 52003, -4741887, 4535189795, -25273021529, 1847538284735, -13861021151998879, 70722327129418887, -268605133000589500717, 23799858495522904017785, -108128513649040594935009169, 1403426321169925536031927183, -27867021017469762051006316943497
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OFFSET
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0,4
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LINKS
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EXAMPLE
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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)).
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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