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Signed denominators of the reduced form of the coefficients of degree 2n terms of the Maclaurin series of (t/sinh(t))^x in t.
0

%I #26 Feb 15 2019 08:47:41

%S 1,-6,360,-45360,5443200,-359251200,5884534656000,-35307207936000,

%T 144053408378880000,-1034591578977116160000,3414152210624483328000000,

%U -471153005066178699264000000,15434972445968014187888640000000,-926009834675808085127331840000000,161141112335906068121557401600000000,-6923589032624540122910835317145600000000,56496486506216247402952416187908096000000000

%N 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 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 C.-K. Fok, <a href="http://arxiv.org/abs/1510.01984">Adams operations on classical compact Lie groups</a>, preprint.

%e p_n(x):=coefficient of t^{2n} of the Maclaurin series of (t/sinh(t))^x

%e p_0(x)=1

%e p_1(x)=-x/6

%e p_2(x)=x(5x+2)/360

%e p_3(x)=-(35x^3+42x^2+16x)/45360

%e p_4(x)=175x^4+420x^3+404x^2+144x/5443200

%e p_5(x)=-(385x^5+1540x^4+2684x^3+2288x^2+768x)/359251200

%t a[n_] := Module[{c}, c = SeriesCoefficient[(t/Sinh[t])^x, {t, 0, 2(n-1)}] // Together; Sign[Numerator[c /. x -> 1]] Denominator[c]];

%t Table[a[n], {n, 1, 17}] (* Updated by _Jean-François Alcover_, Feb 15 2019 *)

%Y Cf. A202367.

%K sign

%O 1,2

%A _Chi-Kwong Fok_, Sep 14 2015

%E Sign added