A143088 - OEIS

%I #12 Sep 13 2024 08:06:46

%S 1,3,3,7,9,7,15,21,21,15,31,45,49,45,31,63,93,105,105,93,63,127,189,

%T 217,225,217,189,127,255,381,441,465,465,441,381,255,511,765,889,945,

%U 961,945,889,765,511,1023,1533,1785,1905,1953,1953,1905,1785,1533,1023,2047

%N Triangle T(n,m) = (2^(m+1) - 1) * (2^(n-m+1) - 1), read by rows, 0 <= m <= n.

%C Row sums are A045618.

%C Considered as a square array A(m,n) = (2^m - 1)(2^n - 1), (m, n >= 1), read by rising antidiagonals, this gives the number of m X n matrices of rank 1 over the field F_2. For a different field F_q, that number would be A(m,n) = (q^m - 1)(q^n - 1)/(q - 1). It satisfies the recurrence relation A(m,n) = A(m,n-1)*q + A(m,1). - _M. F. Hasler_, Sep 12 2024

%F T(n,m) = T(n,n-m).

%F T(n,0) = T(n,n) = 2^(n+1) - 1. - _M. F. Hasler_, Sep 12 2024

%e 1;

%e 3, 3;

%e 7, 9, 7;

%e 15, 21, 21, 15;

%e 31, 45, 49, 45, 31;

%e 63, 93, 105, 105, 93, 63;

%e 127, 189, 217, 225, 217, 189, 127;

%e 255, 381, 441, 465, 465, 441, 381, 255;

%e 511, 765, 889, 945, 961, 945, 889, 765, 511;

%e 1023, 1533, 1785, 1905, 1953, 1953, 1905, 1785, 1533, 1023;

%e 2047, 3069, 3577, 3825, 3937, 3969, 3937, 3825, 3577, 3069, 2047;

%e ...

%e From _M. F. Hasler_, Sep 12 2024: (Start)

%e Considered as a square array A(m,n), read by antidiagonals, with m, n >= 1, this represents the following matrix A:

%e     m \ n: 1  |  2  |  3  |  4  |  5  | ...

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

%e     1  |   1  |  3  |   7 |  15 |  31 | ...

%e     2  |   3  |  9  |  21 |  45 |  93 | ...

%e     3  |   7  | 21  |  49 | 105 | 217 | ...

%e     4  |  15  | 45  | 105 | 225 | 465 | ...

%e    ...

%e Here each row equals twice the previous row plus the first row, and likewise for columns. See my comment relating this to rank 1 matrices over F_2. (End)

%t Table[Table[(2^(m + 1) - 1)*(2^(n - m + 1) - 1), {m, 0, n}], {n, 0, 10}]; Flatten[%]

%o (PARI) T(n,m) = (2^(m+1) - 1) * (2^(n-m+1) - 1) \\ _M. F. Hasler_, Sep 12 2024

%Y Cf. A000225 (first column), A068156 (second column).

%K nonn,tabl

%O 0,2

%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 16 2008