A377065 -id:A377065 - cp's OEIS frontend

A377063 Array read by antidiagonals: T(n,k) is the number of {-1,0,1} n X k matrices with all rows and columns summing to zero up to permutations of rows.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 4, 2, 1, 1, 1, 1, 10, 6, 3, 1, 1, 1, 1, 26, 30, 12, 3, 1, 1, 1, 1, 71, 166, 117, 18, 4, 1, 1, 1, 1, 197, 981, 1421, 345, 30, 4, 1, 1, 1, 1, 554, 5937, 20326, 9691, 1042, 42, 5, 1, 1, 1, 1, 1570, 36646, 307063, 336596, 63076, 2746, 63, 5, 1, 1

Offset: 0

Andrew Howroyd, Oct 14 2024

Columns are not permutable.

Equivalently, the number of n X k 0..2 arrays with row sums k and column sums n up to permutations of rows.

			Array begins:
===================================================
n\k | 0 1 2  3    4      5        6           7 ...
----+----------------------------------------------
  0 | 1 1 1  1    1      1        1           1 ...
  1 | 1 1 1  1    1      1        1           1 ...
  2 | 1 1 2  4   10     26       71         197 ...
  3 | 1 1 2  6   30    166      981        5937 ...
  4 | 1 1 3 12  117   1421    20326      307063 ...
  5 | 1 1 3 18  345   9691   336596    12650093 ...
  6 | 1 1 4 30 1042  63076  5328136   506525279 ...
  7 | 1 1 4 42 2746 369036 76292516 18490880339 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..434 (first 29 antidiagonals)

Main diagonal is A377064.
Rows n=0..4 are A000012, A000012, A257520, A377065, A377066.
Columns k=0..4 are A000012, A000012, A008619, A377067, A377068.
Cf. A334549.

A377067 Number of n X 3 0..2 matrices with row sums 3 and column sums n up to permutations of rows.

Original entry on oeis.org

1, 1, 4, 6, 12, 18, 30, 42, 63, 85, 118, 154, 204, 258, 330, 408, 507, 615, 748, 892, 1066, 1254, 1476, 1716, 1995, 2295, 2640, 3010, 3430, 3880, 4386, 4926, 5529, 6171, 6882, 7638, 8470, 9352, 10318, 11340, 12453, 13629, 14904, 16248, 17700, 19228, 20872, 22600, 24453, 26397, 28476

Offset: 0

Andrew Howroyd, Oct 15 2024

Also, the number of n X 3 {-1,0,1} matrices with all rows and columns summing to zero up to permutations of rows.

			The a(2) = 4 matrices are:
  [1 1 1]  [2 1 0]  [2 0 1]  [1 2 0]
  [1 1 1]  [0 1 2]  [0 2 0]  [1 0 2]
The a(3) = 6 matrices are:
  [1 1 1]  [2 1 0]  [2 0 1]  [1 2 0]  [2 1 0]  [2 0 1]
  [1 1 1]  [0 1 2]  [0 2 0]  [1 0 2]  [1 0 2]  [1 2 0]
  [1 1 1]  [1 1 1]  [1 1 1]  [1 1 1]  [0 2 1]  [0 1 2]

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-3,-1,1,3,-1,-2,1).

Column k=3 of A377063.

Cf. A172634, A377065, A377068.

Vec((1 - x + x^2)/((1 - x)^5*(1 + x)^2*(1 + x + x^2)) + O(x^51))

G.f.: (2/(1 - x^3) - 1)/((1 - x)*(1 - x^2)^3).

G.f.: (1 - x + x^2)/((1 - x)^5*(1 + x)^2*(1 + x + x^2)).

A377066 Number of 4 X n 0..2 matrices with row sums n and column sums 4 up to permutations of rows.

Original entry on oeis.org

1, 1, 3, 12, 117, 1421, 20326, 307063, 4809897, 77098437, 1257981093, 20817768368, 348552520988, 5893520355308, 100492937876761, 1726068011602392, 29836176505279377, 518637160845827153, 9060385447950862705, 158987518980922356784, 2801031979220628009327

Offset: 0

Andrew Howroyd, Oct 15 2024

nonn

Also, the number of 4 X n {-1,0,1} matrices with all rows and columns summing to zero up to permutations of rows.

			The a(2) = 3 matrices are:
   [1 1]   [2 0]   [2 0]
   [1 1]   [1 1]   [2 0]
   [1 1]   [1 1]   [0 2]
   [1 1]   [0 2]   [0 2]

Row n=4 of A377063.

Cf. A172642, A377065, A377068.

cp's OEIS Frontend

A377063 Array read by antidiagonals: T(n,k) is the number of {-1,0,1} n X k matrices with all rows and columns summing to zero up to permutations of rows.

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A377067 Number of n X 3 0..2 matrices with row sums 3 and column sums n up to permutations of rows.

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A377066 Number of 4 X n 0..2 matrices with row sums n and column sums 4 up to permutations of rows.

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