A343874 Array read by antidiagonals: T(n,k) is the number of n X n nonnegative integer matrices with sum of elements equal to k, up to rotational symmetry.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 3, 3, 1, 0, 1, 5, 13, 4, 1, 0, 1, 10, 43, 36, 7, 1, 0, 1, 14, 129, 204, 85, 9, 1, 0, 1, 22, 327, 980, 735, 171, 13, 1, 0, 1, 30, 761, 3876, 5145, 2109, 313, 16, 1, 0, 1, 43, 1619, 13596, 29715, 20610, 5213, 528, 21, 1

Offset: 0

Author

Andrew Howroyd, May 06 2021

Keywords

• nonn
• tabl

Examples

			Array begins:
=====================================================
n\k | 0  1   2    3     4      5       6        7
----+------------------------------------------------
  0 | 1  0   0    0     0      0       0        0 ...
  1 | 1  1   1    1     1      1       1        1 ...
  2 | 1  1   3    5    10     14      22       30 ...
  3 | 1  3  13   43   129    327     761     1619 ...
  4 | 1  4  36  204   980   3876   13596    42636 ...
  5 | 1  7  85  735  5145  29715  148561   657511 ...
  6 | 1  9 171 2109 20610 164502 1124382  6744582 ...
  7 | 1 13 313 5213 67769 717509 6457529 50732669 ...
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325

Crossrefs

Rows n=0..4 are A000007, A000012, A008610, A054771, A054773.

Columns k=0..1 are A000012, A004652.

Cf. A054772 (binary case), A318795, A343095, A343875.

Programs

PARI

U(n,s)={(s(1)^(n^2) + s(1)^(n%2)*(2*s(4)^(n^2\4) + s(2)^(n^2\2)))/4}
T(n,k)={polcoef(U(n,i->1/(1-x^i) + O(x*x^k)), k)}