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

Author

Nathaniel Johnston, Sep 11 2014

Keywords

• nonn
• tabl

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).

Examples

			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, ...
		

Links

• 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.

Crossrefs

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