login
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.
6

%I #9 Jan 02 2022 18:29:49

%S 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 129,204,85,9,1,0,1,22,327,980,735,171,13,1,0,1,30,761,3876,5145,2109,

%U 313,16,1,0,1,43,1619,13596,29715,20610,5213,528,21,1

%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 rotational symmetry.

%H Andrew Howroyd, <a href="/A343874/b343874.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 5 10 14 22 30 ...

%e 3 | 1 3 13 43 129 327 761 1619 ...

%e 4 | 1 4 36 204 980 3876 13596 42636 ...

%e 5 | 1 7 85 735 5145 29715 148561 657511 ...

%e 6 | 1 9 171 2109 20610 164502 1124382 6744582 ...

%e 7 | 1 13 313 5213 67769 717509 6457529 50732669 ...

%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)))/4}

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

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

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

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

%K nonn,tabl

%O 0,13

%A _Andrew Howroyd_, May 06 2021