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A262179 Signed denominators of the reduced form of the coefficients of degree 2n terms of the Maclaurin series of (t/sinh(t))^x in t.

%I A262179 #26 Feb 15 2019 08:47:41
%S A262179 1,-6,360,-45360,5443200,-359251200,5884534656000,-35307207936000,
%T A262179 144053408378880000,-1034591578977116160000,3414152210624483328000000,
%U A262179 -471153005066178699264000000,15434972445968014187888640000000,-926009834675808085127331840000000,161141112335906068121557401600000000,-6923589032624540122910835317145600000000,56496486506216247402952416187908096000000000
%N A262179 Signed denominators of the reduced form of the coefficients of degree 2n terms of the Maclaurin series of (t/sinh(t))^x in t.
%C A262179 Conjecture: this is also the integer sequence A202367 up to sign. These numbers show up in the formula for eigenvectors of Adams operations on the K-theory of unitary groups.
%H A262179 C.-K. Fok, <a href="http://arxiv.org/abs/1510.01984">Adams operations on classical compact Lie groups</a>, preprint.
%e A262179 p_n(x):=coefficient of t^{2n} of the Maclaurin series of (t/sinh(t))^x
%e A262179 p_0(x)=1
%e A262179 p_1(x)=-x/6
%e A262179 p_2(x)=x(5x+2)/360
%e A262179 p_3(x)=-(35x^3+42x^2+16x)/45360
%e A262179 p_4(x)=175x^4+420x^3+404x^2+144x/5443200
%e A262179 p_5(x)=-(385x^5+1540x^4+2684x^3+2288x^2+768x)/359251200
%t A262179 a[n_] := Module[{c}, c = SeriesCoefficient[(t/Sinh[t])^x, {t, 0, 2(n-1)}] // Together; Sign[Numerator[c /. x -> 1]] Denominator[c]];
%t A262179 Table[a[n], {n, 1, 17}] (* Updated by _Jean-François Alcover_, Feb 15 2019 *)
%Y A262179 Cf. A202367.
%K A262179 sign
%O A262179 1,2
%A A262179 _Chi-Kwong Fok_, Sep 14 2015
%E A262179 Sign added