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 A164655 #28 Mar 31 2024 14:05:36
%S A164655 1,28,3527,1213136,32797547,43684790932,96017087247229,96044168328256,
%T A164655 471956397645187853,3237597973008257555852,462561506842656976961,
%U A164655 5628425850334528955928112,703596058798919360293439483,18998011529681231695738912916,463360571051954739540899597748949
%N A164655 Numerators of partial sums of Theta(3) = Sum_{j>=1} 1/(2*j-1)^3.
%C A164655 Warning: Usually, Theta3(x) = Sum_{n=-oo..+oo} x^(n^2). - _Joerg Arndt_, Mar 31 2024
%C A164655 The denominators look like those given for the partial sums of another series in A128507.
%C A164655 Rationals (partial sums) Theta(3,n) := Sum_{j=1..n} 1/(2*j-1)^3 (in lowest terms). The limit of these rationals is Theta(3) = (1-1/2^3)*Zeta(3) approximately 1.051799790 (Zeta(n) is the Euler-Riemann zeta function).
%C A164655 This is a member of the k-family of rational sequences Theta(k,n) := Sum_{j=1..n} 1/(2*j-1)^k, k >= 1, which coincides for k=1 with A025550/A025547 (but only for the first 38 terms), for k=2 with A120268/A128492, for k=3 with a(n)/A128507(n) (the denominators may depart for higher n values), A120269/A128493 and A164656/A164657, for k=4 and 5, respectively.
%H A164655 R. Ayoub, <a href="http://www.jstor.org/stable/2319041">Euler and the Zeta Function</a>, Am. Math. Monthly 81 (1974) 1067-1086, p. 1070.
%H A164655 Wolfdieter Lang, <a href="/A164655/a164655.pdf">Theta(k, n), k-family of rational sequences and limits</a>.
%F A164655 a(n) = numerator(Theta(3,n)) = numerator(Sum_{j=1..n} 1/(2*j-1)^3), n >= 1.
%F A164655 Theta(3,n) = (-Psi(2, 1/2) + Psi(2, n+1/2))/16, n >= 1, where Psi(n, k) = Polygamma(n,k) is the n-th derivative of the digamma function. Psi(2, 1/2) = -14*Zeta(3). - _Jean-François Alcover_, Dec 02 2013
%e A164655 Rationals Theta(3,n): [1, 28/27, 3527/3375, 1213136/1157625, 32797547/31255875, 43684790932/41601569625, ...].
%t A164655 r[n_] := Sum[1/(2*j-1)^3, {j, 1, n}]; (* or r[n_] := (PolyGamma[2, n+1/2] - PolyGamma[2, 1/2])/16 // FullSimplify; *) Table[r[n] // Numerator, {n, 1, 15}] (* _Jean-François Alcover_, Dec 02 2013 *)
%K A164655 nonn,frac,easy
%O A164655 1,2
%A A164655 _Wolfdieter Lang_, Oct 16 2009