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%I #25 Mar 01 2022 07:25:58
%S 1,39,241,34883,14039,1516871,7601151,875425657,7887002813,
%T 7095769757767,14199583385459,75087685321529,75113436870869,
%U 927229349730873529,927436191807263569,305182576081725442901,23479178371879154033,37713848011377144613,37717984058802320713,135759786815564675620247
%N Numerator of partial sums of a certain series.
%C The denominators are given in A101628.
%C Second member (m=3) of a family defined in A101028.
%C The limit s=lim(s(n),n->infinity) with the s(n) defined below equals 8*sum(zeta(2*k+1)/3^(2*k),k=1..infinity) with Euler's (or Riemann's) zeta function. This limit is 12*(log(3)-1) = 1.18334746...; see the Abramowitz-Stegun (given in A101028) reference p. 259, eq. 6.3.15 with z=1/3 together with p. 258.
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H Wolfdieter Lang, <a href="/A101627/a101627.txt">Rationals s(n) and more.</a>
%F a(n)=numerator(s(n)) with s(n)=24*sum(1/((3*k-1)*(3*k)*(3*k+1)), k=1..n).
%e s(3)= 24*(1/(2*3*4)+ 1/(5*6*7) + 1/(8*9*10)) = 241/210, hence a(3)=241 and A101628(3)=210.
%t Numerator[Accumulate[Table[8/(9k^3-k),{k,20}]]]
%o (PARI) a(n) = numerator(24*sum(k=1, n, 1/((3*k-1)*(3*k)*(3*k+1))));
%Y Cf. A101028 (m=2), A101629 (m=4), A101631 (m=5).
%Y Cf. A101628 (denominators).
%K nonn,frac,easy
%O 1,2
%A _Wolfdieter Lang_, Dec 23 2004
%E More terms from _Michel Marcus_, Mar 01 2022