cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A334549 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.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 7, 7, 1, 1, 1, 1, 19, 31, 19, 1, 1, 1, 1, 51, 175, 175, 51, 1, 1, 1, 1, 141, 991, 2371, 991, 141, 1, 1, 1, 1, 393, 5881, 32611, 32611, 5881, 393, 1, 1, 1, 1, 1107, 35617, 481381, 1084851, 481381, 35617, 1107, 1, 1
Offset: 0

Views

Author

Andrew Howroyd, May 09 2020

Keywords

Comments

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

Examples

			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   3     7      19         51          141          393 ...
  3 | 1 1   7    31     175        991         5881        35617 ...
  4 | 1 1  19   175    2371      32611       481381      7343449 ...
  5 | 1 1  51   991   32611    1084851     39612501   1509893001 ...
  6 | 1 1 141  5881  481381   39612501   3680774301 360255871641 ...
  7 | 1 1 393 35617 7343449 1509893001 360255871641 ...
     ...
The T(3,2) = 7 matrices are:
  [0 0]  [ 0  0]  [ 0  0]  [ 1 -1]  [-1  1]  [ 1 -1]  [-1  1]
  [0 0]  [ 1 -1]  [-1  1]  [ 0  0]  [ 0  0]  [-1  1]  [ 1 -1]
  [0 0]  [-1  1]  [ 1 -1]  [-1  1]  [ 1 -1]  [ 0  0]  [ 0  0]

Links

Crossrefs

Columns k=0..14 are A000012, A000012, A002426, A172634, A172642, A172639, A172633, A172636, A172638, A172641, A172637, A172644, A172640, A172643, A172635.
Main diagonal is A172645.
Cf. A008300, A333901, A376935, A377063 (up to row permutations).

Formula

T(n,k) = T(k,n).

A377065 Number of 3 X n 0..2 matrices with row sums n and column sums 3 up to permutations of rows.

Original entry on oeis.org

1, 1, 2, 6, 30, 166, 981, 5937, 36646, 229350, 1451757, 9274057, 59699729, 386798777, 2520034050, 16497343046, 108454221206, 715629888822, 4737625385061, 31456633327905, 209418369288865, 1397521222483385, 9346484009527370, 62632803958053870, 420481623373564025
Offset: 0

Views

Author

Andrew Howroyd, Oct 15 2024

Keywords

Comments

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

Examples

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

Links

Crossrefs

Row n=3 of A377063.
Cf. A172634, A377066, A377067.

Programs

  • PARI
    a(n)={(5+sum(i=0, n, sum(j=0, i, (-1)^(n-i)*binomial(n, i)*binomial(i, j)^3)))/6}

Formula

a(n) = (A172634(n) - 1)/6 + 1.
a(n) = (5 + Sum_{i=0..n} Sum_{j=0..i} (-1)^(n-i)*binomial(n, i)*binomial(i, j)^3)/6.

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

Views

Author

Andrew Howroyd, Oct 15 2024

Keywords

Comments

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

Examples

			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]

Links

Crossrefs

Column k=3 of A377063.
Cf. A172634, A377065, A377068.

Programs

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

Formula

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

Views

Author

Andrew Howroyd, Oct 15 2024

Keywords

Comments

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

Examples

			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]

Crossrefs

Row n=4 of A377063.
Cf. A172642, A377065, A377068.

A377068 Number of n X 4 0..2 matrices with row sums 4 and column sums n up to permutations of rows.

Original entry on oeis.org

1, 1, 10, 30, 117, 345, 1042, 2746, 7063, 16759, 38440, 83476, 175473, 354105, 694278, 1318222, 2440888, 4402852, 7770908, 13418156, 22734824, 37807500, 61839510, 99526422, 157858478, 246864782, 381087734, 580990046, 875572808, 1304930876, 1924761300, 2810843268
Offset: 0

Views

Author

Andrew Howroyd, Oct 15 2024

Keywords

Comments

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

Examples

			The a(2) = 10 matrices are:
  [1 1 2 0]  [1 2 1 0]  [1 2 0 1]  [2 1 1 0]  [2 1 0 1]  [2 0 1 1]
  [1 1 0 2]  [1 0 1 2]  [1 0 2 1]  [0 1 1 2]  [0 1 2 1]  [0 2 1 1]
.
  [1 1 1 1]  [2 2 0 0]  [2 0 2 0]  [2 0 0 2]
  [1 1 1 1]  [0 0 2 2]  [0 2 0 2]  [0 2 2 0]

Crossrefs

Column k=4 of A377063.
Cf. A172642, A377066, A377067.

A377064 Number of {-1,0,1} n X n matrices with all rows and columns summing to zero up to permutations of rows.

Original entry on oeis.org

1, 1, 2, 6, 117, 9691, 5328136, 18490880339, 437425741017623, 72201603445260460729, 85704032961379965273243136, 746789021667791689307771100458717, 48671404211097237572497575028382156068182, 24090982261278741928086824237920135192809826750606, 91765357087196227652413496510903409610512369965590600589002
Offset: 0

Views

Author

Andrew Howroyd, Oct 15 2024

Keywords

Comments

Columns are not permutable.
Equivalently, the number of n X n 0..2 arrays with row sums n and column sums n up to permutation of rows.

Crossrefs

Main diagonal of A377063.
Cf. A172645.
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.