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Denominator of Sum_{k=1..n} (-1)^(k+1)/k^3.
4

%I #35 Jul 13 2023 16:29:06

%S 1,8,216,1728,216000,216000,74088000,592704000,16003008000,

%T 16003008000,21300003648000,21300003648000,46796108014656000,

%U 6685158287808000,6685158287808000,53481266302464000,262753461344005632000,262753461344005632000

%N Denominator of Sum_{k=1..n} (-1)^(k+1)/k^3.

%C For n = 1 to n = 13, a(n) = A195506(n), but a(14) = 6685158287808000 <> 46796108014656000 = A195506(14).

%C Lim_{n -> infinity} A136675(n)/a(n) = A197070.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Dirichlet_eta_function">Dirichlet eta function</a>.

%e The first few fractions are 1, 7/8, 197/216, 1549/1728, 195353/216000, 194353/216000, 66879079/74088000, 533875007/592704000, ... = A136675/A334582.

%p b := proc(n) local k: add((-1)^(k + 1)/k^3, k = 1 .. n): end proc:

%p seq(denom(b(n)), n=1..30);

%t Denominator @ Accumulate[Table[(-1)^(k + 1)/k^3, {k, 1, 18}]] (* _Amiram Eldar_, May 08 2020 *)

%o (PARI) a(n) = denominator(sum(k=1, n, (-1)^(k+1)/k^3)); \\ _Michel Marcus_, May 07 2020

%Y Cf. A136675 (numerators), A195506, A197070.

%K nonn,frac

%O 1,2

%A _Petros Hadjicostas_, May 06 2020

%E Offset changed to 1 by _Georg Fischer_, Jul 13 2023