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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.

%I A343874 #9 Jan 02 2022 18:29:49
%S A343874 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,
%T A343874 129,204,85,9,1,0,1,22,327,980,735,171,13,1,0,1,30,761,3876,5145,2109,
%U A343874 313,16,1,0,1,43,1619,13596,29715,20610,5213,528,21,1
%N 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.
%H A343874 Andrew Howroyd, <a href="/A343874/b343874.txt">Table of n, a(n) for n = 0..1325</a>
%e A343874 Array begins:
%e A343874 =====================================================
%e A343874 n\k | 0  1   2    3     4      5       6        7
%e A343874 ----+------------------------------------------------
%e A343874   0 | 1  0   0    0     0      0       0        0 ...
%e A343874   1 | 1  1   1    1     1      1       1        1 ...
%e A343874   2 | 1  1   3    5    10     14      22       30 ...
%e A343874   3 | 1  3  13   43   129    327     761     1619 ...
%e A343874   4 | 1  4  36  204   980   3876   13596    42636 ...
%e A343874   5 | 1  7  85  735  5145  29715  148561   657511 ...
%e A343874   6 | 1  9 171 2109 20610 164502 1124382  6744582 ...
%e A343874   7 | 1 13 313 5213 67769 717509 6457529 50732669 ...
%e A343874   ...
%o A343874 (PARI)
%o A343874 U(n,s)={(s(1)^(n^2) + s(1)^(n%2)*(2*s(4)^(n^2\4) + s(2)^(n^2\2)))/4}
%o A343874 T(n,k)={polcoef(U(n,i->1/(1-x^i) + O(x*x^k)), k)}
%Y A343874 Rows n=0..4 are A000007, A000012, A008610, A054771, A054773.
%Y A343874 Columns k=0..1 are A000012, A004652.
%Y A343874 Cf. A054772 (binary case), A318795, A343095, A343875.
%K A343874 nonn,tabl
%O A343874 0,13
%A A343874 _Andrew Howroyd_, May 06 2021