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 1-9 of 9 results.

A084546 Triangle read by rows: T(n,k) = C( C(n,2), k) for n >= 0, 0 <= k <= C(n,2).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 1, 1, 6, 15, 20, 15, 6, 1, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 1, 15, 105, 455, 1365, 3003, 5005, 6435, 6435, 5005, 3003, 1365, 455, 105, 15, 1, 1, 21, 210, 1330, 5985, 20349, 54264, 116280, 203490, 293930, 352716, 352716, 293930, 203490, 116280, 54264, 20349, 5985, 1330, 210, 21, 1
Offset: 0

Views

Author

N. J. A. Sloane, Jul 13 2003

Keywords

Comments

T(n,k) gives number of labeled simple graphs with n nodes and k edges.

Examples

			Triangle begins:
  1;
  1;
  1, 1;
  1, 3,  3,  1;
  1, 6, 15, 20, 15, 6, 1;
  ...

References

  • J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 517.

Links

Crossrefs

Cf. A083029. A subset of the rows of Pascal's triangle A007318.
Cf. A006125 (row sums), A008406 (unlabeled graphs).
Main diagonal gives A116508.

Programs

  • Maple
    C:= binomial:
T:= (n, k)-> C( C(n, 2), k):
seq(seq(T(n, k), k=0..C(n, 2)), n=0..10);  # Alois P. Heinz, Feb 17 2023
  • Mathematica
    Table[Table[Binomial[Binomial[n,2],k],{k,0,Binomial[n,2]}],{n,1,7}]//Grid (* Geoffrey Critzer, Apr 28 2011 *)

Extensions

T(0,0)=1 prepended by Alois P. Heinz, Feb 17 2023

A085656 Number of positive-definite real {0,1} n X n matrices.

Original entry on oeis.org

1, 3, 27, 681, 43369, 6184475, 1688686483, 665444089745
Offset: 1

Views

Author

N. J. A. Sloane, Jul 12 2003

Keywords

Comments

A real matrix M is positive-definite if x M x' > 0 for all nonzero real vectors x. Equivalently, all eigenvalues of M + M' are positive.
M need not be symmetric. For the number of different values of M + M' see A085657. - Max Alekseyev, Dec 13 2005

Examples

			For n = 2 the three matrices are {{{1, 0}, {0, 1}}, {{1, 0}, {1, 1}}, {{1, 1}, {0, 1}}}.

Links

Crossrefs

Cf. A055165, which counts nonsingular {0, 1} matrices and A085506, which counts {-1, 0, 1} matrices with positive eigenvalues.
Cf. A085657, A085658, A086215, A038379 (positive semi-definite matrices), A080858, A083029.

Programs

  • Mathematica
    Table[Count[Tuples[{0, 1}, {n, n}], ?PositiveDefiniteMatrixQ], {n, 4}] (* _Eric W. Weisstein, Jan 03 2021 *)
  • PARI
    { a(n) = M=matrix(n,n,i,j,2*(i==j)); r=0; b(1); r } { b(k) = local(t); if(k>n, t=0; for(i=1,n, for(j=1,i-1, if(M[i,j]==1,t++); )); r+=2^t; return; ); forvec(x=vector(k-1,i,[0,1]), for(i=1,k-1,M[k,i]=M[i,k]=x[i]); if( matdet(vecextract(M,2^k-1,2^k-1),1)>0, b(k+1) ) ) } (Alekseyev)

Extensions

More terms from Max Alekseyev, Dec 13 2005

A085658 Number of n X n symmetric positive semi-definite matrices with 2's on the main diagonal and 1's and 0's elsewhere.

Original entry on oeis.org

1, 2, 8, 64, 924, 21737, 749684, 33568376
Offset: 1

Views

Author

N. J. A. Sloane, Jul 12 2003

Keywords

Comments

Of course the total number of symmetric matrices of this type (not necessarily positive semi-definite) is 2^C(n,2).
This gives the number of different values of M + M' where M runs through the matrices counted in A038379. - Max Alekseyev, Nov 11 2006

Examples

			The matrix
2 0 0 0 1
0 2 0 1 1
0 0 2 1 1
0 1 1 2 0
1 1 1 0 2
is one of the 100 5 X 5 matrices which are not positive semi-definite.
Its eigenvalues are approximately [2., -0.135779205069857, 4.135779205069857, 1.337846553138044, 2.662153446861956]

Links

Crossrefs

Cf. A085658, A080858, A083029.

Extensions

3 more terms from Max Alekseyev, Nov 08 2006

A085657 Number of n X n symmetric positive definite matrices with 2's on the main diagonal and 1's and 0's elsewhere.

Original entry on oeis.org

1, 2, 8, 61, 819, 17417, 506609, 15582436
Offset: 1

Views

Author

N. J. A. Sloane, Jul 12 2003

Keywords

Comments

Of course the total number of symmetric matrices of this type (not necessarily positive definite) is 2^C(n,2).
This gives the number of different values of A + A' where A runs through the matrices counted in A085656. - Max Alekseyev, Dec 13 2005

Examples

			The singular matrix
2 0 1 1
0 2 1 1
1 1 2 0
1 1 0 2
is one of the three 4 X 4 matrices which are not positive definite.

Links

Crossrefs

Cf. A085659, A080858, A083029.
Cf. A085656, A114601.

Programs

  • PARI
    { a(n) = M=matrix(n,n,i,j,2*(i==j)); r=0; b(1); r } { b(k) = if(k>n, r++; return); forvec(x=vector(k-1,i,[0,1]), for(i=1,k-1,M[k,i]=M[i,k]=x[i]); if( matdet(vecextract(M,2^k-1,2^k-1),1)>0, b(k+1) ) ) } (Max Alekseyev)

Extensions

More terms from Max Alekseyev, Dec 13 2005

A038379 Number of real {0,1} n X n matrices A such that M = A + A' has 2's on the main diagonal, 0's and 1's elsewhere and is positive semi-definite.

Original entry on oeis.org

1, 3, 27, 729, 52649, 9058475, 3383769523, 2520512534065
Offset: 1

Views

Author

N. J. A. Sloane, Jul 13 2003

Keywords

Comments

Necessarily A has all 1's on the main diagonal.
A real matrix M is positive semi-definite if its eigenvalues are >= 0.
For n <= 4, a(n) equals the upper bound 3^C(n,2).
For the number of different values of symmetric parts A + A', see A085658. - Max Alekseyev, Nov 11 2006

Links

Crossrefs

Cf. A055165, which counts nonsingular {0, 1} matrices, A003024, which counts {0, 1} matrices with positive eigenvalues, A085656 (positive definite matrices).
Cf. A085657, A085658, A080858, A083029.

Formula

a(n) = Sum_{k=0..C(n,2)} 2^k * A083029(n,k).

Extensions

Definition corrected Nov 10 2006
a(6)-a(8) from Max Alekseyev, Nov 11 2006
Edited by Max Alekseyev, Jun 05 2024

A084552 Number of n X n symmetric positive definite matrices with 2's on the main diagonal and -1's and 0's elsewhere.

Original entry on oeis.org

1, 2, 7, 38, 286, 2686, 28512, 312572, 3337588, 40963216
Offset: 1

Views

Author

N. J. A. Sloane, Jul 13 2003

Keywords

Comments

Of course the total number of symmetric matrices of this type (not necessarily positive definite) is 2^C(n,2).
This gives the number of different values of M + M' where M runs through the matrices counted in A127502.

Examples

			The singular matrix
2 -1 -1
-1 2 -1
-1 -1 2
is the only 3 X 3 matrix of this type which is not positive definite.

Links

Crossrefs

Cf. A084553, A085659, A080858, A083029.

Extensions

a(6)-a(10) from Max Alekseyev, Jan 16 2006

A084553 Number of n X n symmetric positive semi-definite matrices with 2's on the main diagonal and -1's and 0's elsewhere.

Original entry on oeis.org

1, 2, 8, 45, 338, 3178, 34346, 396659, 4694705
Offset: 1

Views

Author

N. J. A. Sloane, Jul 13 2003

Keywords

Comments

Of course the total number of symmetric matrices of this type (not necessarily positive definite) is 2^C(n,2).
This gives the number of different values of A + A' where A runs through the matrices counted in A127503.

Examples

			The 5 X 5 matrix
2 -1 -1 -1 -1
-1 2 -1 -1 -1
-1 -1 2 -1 -1
-1 -1 -1 2 -1
-1 -1 -1 -1 2
with eigenvalues -2, 3, 3, 3, 3 is an example of one which is not positive semi-definite.

Links

Crossrefs

Cf. A084552, A085658, A080858, A083029.

Extensions

a(6)-a(9) from Max Alekseyev, Jan 16 2006

A127502 Number of n X n positive definite matrices with 1's on the main diagonal and -1's and 0's elsewhere.

Original entry on oeis.org

1, 3, 19, 201, 3001, 55291, 1115003, 21837649, 373215601, 8282131891
Offset: 1

Views

Author

Max Alekseyev, Jan 16 2007

Keywords

Comments

A real matrix M is positive-definite if x M x' > 0 for all nonzero real vectors x. Equivalently, all eigenvalues of M + M' are positive.
M need not be symmetric. For the number of different values of M + M' see A084552.

Examples

			For n = 2 the three matrices are {{{1, 0}, {0, 1}}, {{1, 0}, {-1, 1}}, {{1, -1}, {0, 1}}}.

Crossrefs

Cf. A085656, A085657, A085658, A086215, A038379, A080858, A083029, A127503.

Programs

  • PARI
    { a(n) = M=matrix(n,n,i,j,2*(i==j)); r=0; b(1); r } { b(k) = local(t); if(k> n, t=0; for(i=1,n, for(j=1,i-1, if(M[i,j]==1,t++); )); r+=2^t; return; ); forvec(x=vector(k-1,i,[ -1,0]), for(i=1,k-1,M[k,i]=M[i,k]=x[i]); if( matdet(vecextract(M,2^k-1,2^k-1),1)>0, b(k+1) ) ) } (Alekseyev)

A127503 Number of n X n matrices A with 1's on the main diagonal and -1's and 0's elsewhere such that A + A' has only 0's and -1's off the main diagonal and is positive semi-definite.

Original entry on oeis.org

1, 3, 27, 281, 3945, 70635, 1437555, 30357425, 628337745
Offset: 1

Views

Author

Max Alekseyev, Jan 16 2007

Keywords

Comments

For number of different values of A + A' see A084553.

Crossrefs

Cf. A085657, A085658, A080858, A083029, A038379, A127502.
Showing 1-9 of 9 results.

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

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