A342319 - OEIS

A342319

a(n) = denominator(((i^n * PolyLog(1 - n, -i) + (-i)^n * PolyLog(1 - n, i))) / (4^n - 2^n)) if n > 0 and a(0) = 1. Here i denotes the imaginary unit.

%I #8 Mar 23 2021 04:26:30

%S 1,2,12,56,120,992,252,16256,240,261632,132,4192256,32760,67100672,12,

%T 1073709056,8160,17179738112,14364,274877382656,6600,4398044413952,

%U 276,70368735789056,65520,1125899873288192,12,18014398375264256,3480,288230375614840832

%N a(n) = denominator(((i^n * PolyLog(1 - n, -i) + (-i)^n * PolyLog(1 - n, i))) / (4^n - 2^n)) if n > 0 and a(0) = 1. Here i denotes the imaginary unit.

%C For comments and references see A342318.

%F a(2*n) = A006953(n).

%F a(2*n+1) = A193475(n).

%e r(n) = 1, 1/2, 1/12, 1/56, 1/120, 5/992, 1/252, 61/16256, 1/240, 1385/261632, 1/132, ...

%p a := n -> `if`(n = 0, 1, `if`(n::even, denom(abs(bernoulli(n))/n), 4^n - 2^n)):

%p seq(a(n), n=0..29);

%t r[s_] := If[s == 0, 1, (I^s PolyLog[1 - s, -I] + (-I)^s PolyLog[1 - s, I]) / (4^s - 2^s)]; Table[r[n], {n, 0, 29}] // Denominator

%Y Cf. A342318 (numerator), A006953, A193475.

%K nonn,frac

%O 0,2

%A _Peter Luschny_, Mar 22 2021