A247306 Array T(n,k) of the expected value of trace(O)^(2k), where O is an n X n orthogonal matrix randomly selected according to Haar measure, read by antidiagonals.

1, 1, 1, 1, 3, 1, 1, 3, 10, 1, 1, 3, 15, 35, 1, 1, 3, 15, 91, 126, 1, 1, 3, 15, 105, 603, 462, 1, 1, 3, 15, 105, 903, 4213, 1716, 1, 1, 3, 15, 105, 945, 8778, 30537, 6435, 1, 1, 3, 15, 105, 945, 10263, 92235, 227475, 24310, 1

Offset

1,5

Comments

The expected value of trace(O)^k, when k is an odd natural number, is zero.

The first 5 rows of this array are A000012, A001700, A099251, A246860, and A247304.

T(n,k) = A001147(k) when k <= n. When k > n, T(n,k) < A001147(k).

Links

Table of n, a(n) for n=1..55.

L. Pastur and V. Vasilchuk, On the moments of traces of matrices of classical groups, Commun. Math. Phys. 252:149-166, 2004.

MathOverflow, Moments of the trace of orthogonal matrices.

Example

The upper-left corner of the array begins:
1, 1,  1,   1,   1,     1,      1, ...
1, 3, 10,  35, 126,   462,   1716, ...
1, 3, 15,  91, 603,  4213,  30537, ...
1, 3, 15, 105, 903,  8778,  92235, ...
1, 3, 15, 105, 945, 10263, 127699, ...

Crossrefs

Cf. A001147, A001700, A099251, A246860, and A247304.

Sequence in context: A024737 A024958 A014516 * A016563 A025254 A245537

Adjacent sequences: A247303 A247304 A247305 * A247307 A247308 A247309

Keyword

nonn,tabl

Author

Nathaniel Johnston, Sep 11 2014

Status

approved