A333166 -id:A333166 - cp's OEIS frontend

A333161 Triangle read by rows: T(n,k) is the number of k-regular graphs on n unlabeled nodes with half-edges.

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 3, 3, 1, 1, 3, 4, 4, 3, 1, 1, 4, 8, 12, 8, 4, 1, 1, 4, 10, 24, 24, 10, 4, 1, 1, 5, 17, 70, 118, 70, 17, 5, 1, 1, 5, 24, 172, 634, 634, 172, 24, 5, 1, 1, 6, 36, 525, 4428, 9638, 4428, 525, 36, 6, 1, 1, 6, 50, 1530, 35500, 187990, 187990, 35500, 1530, 50, 6, 1

Offset: 0

Andrew Howroyd, Mar 11 2020

A half-edge is like a loop except it only adds 1 to the degree of its vertex.
T(n,k) is the number of non-isomorphic n X n symmetric binary matrices with k ones in every row and column and isomorphism being up to simultaneous permutation of rows and columns. The case that allows independent permutations of rows and columns is covered by A333159.
T(n,k) is the number of simple graphs on n unlabeled vertices with every vertex degree being either k or k-1.

			Triangle begins:
  1;
  1, 1;
  1, 2,  1;
  1, 2,  2,   1;
  1, 3,  3,   3,    1;
  1, 3,  4,   4,    3,    1;
  1, 4,  8,  12,    8,    4,    1;
  1, 4, 10,  24,   24,   10,    4,   1;
  1, 5, 17,  70,  118,   70,   17,   5,  1;
  1, 5, 24, 172,  634,  634,  172,  24,  5, 1;
  1, 6, 36, 525, 4428, 9638, 4428, 525, 36, 6, 1;
  ...
The a(2,1) = 2 adjacency matrices are:
  [0 1]  [1 0]
  [1 0]  [0 1]
.
The A(4,2) = 3 adjacency matrices are:
  [0 0 1 1]   [1 1 0 0]   [1 1 0 0]
  [0 0 1 1]   [1 1 0 0]   [1 0 1 0]
  [1 1 0 0]   [0 0 1 1]   [0 1 0 1]
  [1 1 0 0]   [0 0 1 1]   [0 0 1 1]

Andrew Howroyd, Table of n, a(n) for n = 0..230

Columns k=0..3 are A000012, A004526(n+2), A186417, A333163.
Row sums are A333162.
Central coefficients are A333166.
Cf. A051031, A333157, A333159.

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

A333164 Number of 2n X 2n symmetric matrices with entries in {+1, -1} and all rows and columns summing to zero.

Original entry on oeis.org

1, 2, 18, 1760, 1944530, 26615510712, 4762109992158288, 11528251571501588791296, 386860001875783390762182911250, 183238648400953515891207813311894423000, 1242327573587456839123512395835593196519781229768, 121917941188016997420391711911475819481799846311888884541952

Offset: 0

Andrew Howroyd, Mar 11 2020

nonn

a(n) is the number of 2n X 2n symmetric binary matrices with n ones in every row and column.

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

Central coefficients of A333157.

Cf. A213793, A333165, A333166.

a(n) = A333157(2*n, n).

a(n) = A213793(2*n). - Andrew Howroyd, Apr 08 2020

A333165 Number of non-isomorphic 2n X 2n symmetric matrices with entries in {+1, -1} and all rows and columns summing to zero.

Original entry on oeis.org

1, 1, 2, 5, 66, 7937, 10211144, 133506398361, 18551599312980440, 28652629505982770906471, 510824181488832447063505273252

Offset: 0

Andrew Howroyd, Mar 11 2020

a(n) is the number of non-isomorphic 2n X 2n symmetric binary matrices with n ones in every row and column.

			The a(1) = 1 matrix is:
  [+ -]
  [- +]
The a(2) = 2 matrices are:
  [+ + - -]   [+ + - -]
  [+ + - -]   [+ - + -]
  [- - + +]   [- + - +]
  [- - + +]   [- - + +]

Central coefficients of A333159.

Cf. A333164, A333166.

a(n) = A333159(2*n, n).

cp's OEIS Frontend

A333161 Triangle read by rows: T(n,k) is the number of k-regular graphs on n unlabeled nodes with half-edges.

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A333164 Number of 2n X 2n symmetric matrices with entries in {+1, -1} and all rows and columns summing to zero.

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A333165 Number of non-isomorphic 2n X 2n symmetric matrices with entries in {+1, -1} and all rows and columns summing to zero.

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