This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A210244 #32 Feb 16 2025 08:33:17
%S A210244 -1,-1,1,5,-7,-49,-53,2215,1259,-14201,-183197,248885,9583753,
%T A210244 14525053,-554173253,-4573299625,99833187251,215440236599,
%U A210244 -1654012631597,-84480933600305,-36267273557287,10992430255511053,117548575473066241,-1380910044674479865
%N A210244 Numerators of the polylogarithm li(-n,-1/2)/2.
%C A210244 Given an integer n>0, consider the infinite series s(n) = li(-n,-1/2) = Sum_{k>=1} (-1)^k*k^n/2^k. Then s(n)=2*a(n)/A131137(n+1).
%H A210244 S. Sykora, <a href="http://dx.doi.org/10.3247/SL1Math06.002">Finite and Infinite Sums of the Power Series (k^p)(x^k)</a>, Stan's Library Vol.I, April 2006, updated March 2012. See Eq.(29).
%H A210244 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/Polylogarithm.html">MathWorld: Polylogarithm</a>
%F A210244 Recurrence: s(n+1)=(-1/3)*Sum_{i=0..n} binomial(n+1,i)*s(i), with the starting value of s(0)=2/3.
%e A210244 s(1)=-2/9, s(2)=-2/27, s(3)=+2/27, s(4)=+10/81.
%t A210244 nn = 30; s[0] = 1; Do[s[n+1] = (-1/3) Sum[Binomial[n+1,i] s[i], {i, 0, n}], {n, 0, nn}]; Numerator[Table[s[n], {n, 0, nn}]] (* _T. D. Noe_, Mar 20 2012 *)
%t A210244 Table[PolyLog[-n, -1/2]/2, {n, 30}] (* _T. D. Noe_, Mar 23 2012 *)
%o A210244 (PARI) a(n)=numerator(polylog(-n,-1/2)/2) \\ _Charles R Greathouse IV_, Jul 15 2014
%Y A210244 Denominators: A131137, offset by 1.
%Y A210244 Cf. A212846.
%K A210244 sign,frac
%O A210244 1,4
%A A210244 _Stanislav Sykora_, Mar 19 2012