A247306 -id:A247306 - cp's OEIS frontend

A247304 Expected value of trace(O)^(2n), where O is a 5 X 5 orthogonal matrix randomly selected according to Haar measure.

1, 3, 15, 105, 945, 10263, 127699, 1751685, 25807445, 401449271, 6519160851, 109656966853, 1899821492925, 33757439931675, 613135175204151, 11352879180474205, 213825919244349885, 4089023317563827335, 79269914519539177315, 1555788699527063688453

Offset: 1

Nathaniel Johnston, Sep 11 2014

nonn

The corresponding sequences for 2 X 2, 3 X 3, and 4 X 4 matrices are A001700, A099251, and A246860.

This sequence is a bisection of A095922. - Nathaniel Johnston, Nov 03 2014

MathOverflow, Moments of the trace of orthogonal matrices

Cf. A001700, A095922, A099251, A246860, A247306.

A247304 := proc (n) return (1/2)*integrate(integrate((1-cos(x))*(1-cos(y))*(cos(x)-cos(y))^2*(1+2*cos(x)+2*cos(y))^(2*n), y = 0 .. 2*Pi), x = 0 .. 2*Pi)/Pi^2 end proc; seq(A247304(n), n = 1 .. 20);

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

Original entry on oeis.org

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

Offset: 1

Nathaniel Johnston, Sep 05 2014

nonn

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

MathOverflow, Moments of the trace of orthogonal matrices

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

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);

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 *)

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

cp's OEIS Frontend

A247304 Expected value of trace(O)^(2n), where O is a 5 X 5 orthogonal matrix randomly selected according to Haar measure.

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