A246860 Expected value of trace(O)^(2n), where O is a 4 X 4 orthogonal matrix randomly selected according to Haar measure.

1, 3, 15, 105, 903, 8778, 92235, 1023165, 11821953, 141061206, 1727926291, 21634600078, 275950576450, 3576315994020, 46995014634435, 625082431593285, 8403885851894445, 114069364107664350, 1561609592248119645, 21543838447412548410, 299299110959202973710

Offset: 1

Author

Nathaniel Johnston, Sep 05 2014

Keywords

• nonn

Comments

The corresponding sequences for 2 X 2, 3 X 3, and 5 X 5 matrices are A001700, A099251, and A247304.

a(n) is the number of triangulations with middle chord of an 2n+2-gon modulo the cyclic action. So a(n) = A000108(n)^2 - A000107(A000108(n)-1). The first part A000108(n)^2 means the cartes of two n+2-gons separated by the middle chord, second part is the duplicated joins need to be removed. - Yuchun Ji, Aug 11 2020

Links

• MathOverflow, Moments of the trace of orthogonal matrices

Crossrefs

Cf. A000108, A001700, A099251, A247304, A247306.

Programs

Maple

A246860 := proc (n) return (1/8)*integrate(integrate((cos(x)-cos(y))^2*(2*cos(x)+2*cos(y))^(2*n), y = 0 .. 2*Pi), x = 0 .. 2*Pi)/Pi^2+(1/2)*integrate((1-cos(z)^2)*(2*cos(z))^(2*n), z = 0 .. 2*Pi)/Pi end proc; seq(A246860(n), n = 1 .. 21);

Mathematica

a[n_] := a[n] = (1/8)*Integrate[Integrate[(Cos[x] - Cos[y])^2 * (2 Cos[x] + 2 Cos[y])^(2 n), {y, 0, 2 Pi}], {x, 0, 2 Pi}]/ Pi^2 + (1/2)*Integrate[(1 - Cos[z]^2)*(2 Cos[z])^(2 n), {z, 0, 2 Pi}]/Pi;
Table[Print[n, " ", a[n]]; a[n], {n, 1, 21}] (* Jean-François Alcover, Feb 05 2023 *)

Formula

In the MathOverflow link, Nathaniel Johnston conjectures a(n) = A000108(n)*(A000108(n)+1)/2. - Robert Israel, Jan 17 2020