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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.
5

%I #8 Jan 02 2022 18:31:16

%S 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,

%T 113,55,6,1,0,1,16,198,528,410,99,10,1,0,1,20,440,2003,2710,1091,181,

%U 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

%N 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.

%H Andrew Howroyd, <a href="/A343875/b343875.txt">Table of n, a(n) for n = 0..1325</a>

%e Array begins:

%e =====================================================

%e n\k | 0 1 2 3 4 5 6 7

%e ----+------------------------------------------------

%e 0 | 1 0 0 0 0 0 0 0 ...

%e 1 | 1 1 1 1 1 1 1 1 ...

%e 2 | 1 1 3 4 8 10 16 20 ...

%e 3 | 1 3 11 31 84 198 440 904 ...

%e 4 | 1 3 24 113 528 2003 6968 21593 ...

%e 5 | 1 6 55 410 2710 15233 75258 331063 ...

%e 6 | 1 6 99 1091 10488 82704 563864 3376134 ...

%e 7 | 1 10 181 2722 34399 360676 3235551 25387944 ...

%e ...

%o (PARI)

%o 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}

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

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

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

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

%K nonn,tabl

%O 0,13

%A _Andrew Howroyd_, May 06 2021