A343875 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 rotations and reflections.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 3, 3, 1, 0, 1, 4, 11, 3, 1, 0, 1, 8, 31, 24, 6, 1, 0, 1, 10, 84, 113, 55, 6, 1, 0, 1, 16, 198, 528, 410, 99, 10, 1, 0, 1, 20, 440, 2003, 2710, 1091, 181, 10, 1, 0, 1, 29, 904, 6968, 15233, 10488, 2722, 288, 15, 1, 0, 1, 35, 1766, 21593, 75258, 82704, 34399, 5806, 461, 15, 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    4     8     10      16       20 ...
  3 | 1  3  11   31    84    198     440      904 ...
  4 | 1  3  24  113   528   2003    6968    21593 ...
  5 | 1  6  55  410  2710  15233   75258   331063 ...
  6 | 1  6  99 1091 10488  82704  563864  3376134 ...
  7 | 1 10 181 2722 34399 360676 3235551 25387944 ...
  ...
		

Links

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

Crossrefs

Rows n=0..3 are A000007, A000012, A005232, A054343.

Columns 0..1 are A000012, A008805(n-1).

Cf. A054252 (binary case), A318795, A343097, A343874.

Programs

PARI

U(n,s) = {(s(1)^(n^2) + s(1)^(n%2)*(2*s(4)^(n^2\4) + s(2)^(n^2\2)) + 2*s(1)^n*s(2)^(n*(n-1)/2) + 2*(s(1)^(n%2)*s(2)^(n\2))^n )/8}
T(n,k)={polcoef(U(n,i->1/(1-x^i) + O(x*x^k)), k)}