A260151 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).

1, -1, 1, -11, 65, -223193, 52003, -4741887, 4535189795, -25273021529, 1847538284735, -13861021151998879, 70722327129418887, -268605133000589500717, 23799858495522904017785, -108128513649040594935009169, 1403426321169925536031927183, -27867021017469762051006316943497

Offset

0,4

Links

Table of n, a(n) for n=0..17.

Example

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)).

Mathematica

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 *)

Crossrefs

Sequence in context: A125321 A054490 A126479 * A139611 A154617 A297751

Adjacent sequences: A260148 A260149 A260150 * A260152 A260153 A260154

Keyword

sign

Author

Vladimir Reshetnikov, Nov 09 2015

Extensions

More terms from Vaclav Kotesovec, Nov 20 2015

Status

approved