A001501 -id:A001501 - cp's OEIS frontend

A006714 Number of trivalent bipartite labeled graphs with 2n labeled nodes.

10, 840, 257040, 137260200, 118273755600, 154712104747200, 292311804557572800, 766931112143320924800, 2706462791802644002128000, 12512595130808078973370704000, 74130965352250071944327288640000, 552334353713465817349513210512960000, 5092566798555894395129552704613028960000

Offset: 3

N. J. A. Sloane

nonn

R. C. Read incorrectly has a(7) = 118257539400 and a(8) = 154678050727200 which he calculated by hand. - Sean A. Irvine, Jun 27 2017

R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 3..50
R. C. Read, Letter to N. J. A. Sloane, Feb 04 1971 (gives initial terms of this sequence)

Cf. A001501, A002829, A004109, A246599.

(* b stands for A001501 *) b[n_] := n!^2 Sum[2^(2k-n) 3^(k-n) (3(n-k))!  HypergeometricPFQ[{k-n, k-n}, {3(k-n)/2, 1/2 + 3(k-n)/2}, -9/2]/(k! (n-k)!^2), {k, 0, n}]/6^n;
(* c stands for A246599 *) c[n_] := c[n] = Binomial[2n-1, n] b[n] - Sum[ Binomial[2n-1, 2k] Binomial[2k, k] b[k] c[n-k], {k, 1, n-1}];
a[n_] := a[n] = c[n] + Sum[Binomial[2n-1, 2k-1] c[k] a[n-k], {k, 1, n-1}];
Table[a[n], {n, 3, 20}] (* Jean-François Alcover, Jul 07 2018, after Andrew Howroyd *)

\\ here b(n) is A001501
b(n) = {n!^2 * sum(j=0, n, sum(i=0, n-j, my(k=n-i-j); (j + 3*k)! / (3^i * 36^k * i! * k!^2)) / (j! * (-2)^j))}
seq(n)={my(v=vector(n,n,b(n)*binomial(2*n-1,n)), u=vector(n), s=vector(n)); for(n=1, #u, u[n]=v[n] - sum(k=3, n-3, 2*binomial(2*n-1,2*k)*v[k]*u[n-k]); s[n]=u[n] + sum(k=3, n-3, binomial(2*n-1,2*k-1)*u[k]*s[n-k])); s[3..n]} \\ Andrew Howroyd, May 22 2018

a(n) = A246599(n) + Sum_{k=1..n-1} binomial(2*n-1,2*k-1)*A246599(k)*a(n-k). - Andrew Howroyd, May 22 2018

a(n) ~ 3^(n + 1/2) * n^(3*n) / (sqrt(2) * exp(3*n+2)). - Vaclav Kotesovec, Feb 17 2024

a(7)-a(8) corrected and a(9)-a(12) computed with nauty by Sean A. Irvine, Jun 27 2017

Terms a(13) and beyond from Andrew Howroyd, May 22 2018

A079815 Number of equivalent classes of n X n 0-1 matrices with 3 1's in each row and column.

Original entry on oeis.org

0, 0, 1, 1, 2, 7, 16, 71

Offset: 1

Michael Steyer (m.steyer(AT)osram.de), Feb 20 2003

Matrices are considered to belong to the same equivalent class if they can be transformed into each other by successive permutations of rows or columns.
In general, to transform 2 equivalent matrices into each other, it is necessary to first permute rows, then columns, then rows and so on.
From Brendan McKay, Aug 27 2010: (Start)
A079815 appears on the surface to describe the same objects as A000512, but I don't know where the term "71" comes from.
Also the comment "In general, to transform 2 equivalent matrices into each other, it is necessary to first permute rows, then columns, then rows and so on." is wrong - actually only one permutation of rows and one permutation of columns is enough.
I will guess that this sequence counts matrices in which both the rows and columns are in sorted order. The reason I suspect that is because a common way to make such matrices is to alternately sort the rows and columns until it stabilizes.
The value of a(8) should be checked. (End)

			n=4: every matrix with 3 1's in each row and column can be transformed by permutation of rows (or columns) into {1110,1101,1011,0111}, therefore a(4)=1.

Cf. A001501.

Edited by N. J. A. Sloane, Sep 04 2010

A225623 Number of ways to arrange 2n queens on an n X n chessboard, with no more than 2 queens in each row, column or diagonal.

Original entry on oeis.org

0, 1, 2, 11, 92, 1097, 19448, 477136, 14244856, 537809179, 24194010708, 1317062528249

Offset: 1

Vaclav Kotesovec, Aug 04 2013

This problem is slightly different from A000769 or A219760. In the first example on an 8 x 8 board, the queens c7, d5 and e3 (or queens a2, c5 and e8) are in a line, but such case is allowed. The elementary step can be only [0,1], [1,0] or [1,1], not for example [1,2] or [2,3].

Vaclav Kotesovec, Examples

Cf. A000170, A000769, A001499, A001501, A058528, A075754, A172544, A172541.

Definition clarified by Vaclav Kotesovec, Dec 18 2014

a(10)-a(12) from Martin Ehrenstein, Jan 09 2022

A246599 Number of connected trivalent bipartite labeled graphs with 2n labeled nodes.

Original entry on oeis.org

10, 840, 257040, 137214000, 118248530400, 154686980448000, 292276881344448000, 766864651478365440000, 2706292794907249067520000, 12512021073989410699165440000, 74128448237031250090060032000000, 552320243355746711191770103680000000, 5092467146398443040845772685937408000000

Offset: 3

N. J. A. Sloane, Sep 08 2014

nonn

R. C. Read incorrectly has a(7) = 118237555800 and a(8) = 154652926428000 which he calculated by hand.

R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.

Andrew Howroyd, Table of n, a(n) for n = 3..50
R. C. Read, Letter to N. J. A. Sloane, Feb 04 1971 (gives initial terms of this sequence)

Cf. A001501, A002829, A004109, A006714.

b[n_] := n!^2*Sum[2^(2k-n) 3^(k-n)(3(n-k))!*HypergeometricPFQ[{k-n, k-n}, {3(k-n)/2, 1/2 + 3(k-n)/2}, -9/2]/(k! (n-k )!^2), {k, 0, n}]/6^n;
a[n_] := a[n] = Binomial[2n-1, n] b[n] - Sum[Binomial[2n-1, 2k] Binomial[2 k, k] b[k] a[n-k], {k, 1, n-1}];
Table[a[n], {n, 3, 20}] (* Jean-François Alcover, Jul 07 2018, after Andrew Howroyd *)

\\ here b(n) is A001501
b(n) = {n!^2 * sum(j=0, n, sum(i=0, n-j, my(k=n-i-j); (j + 3*k)! / (3^i * 36^k * i! * k!^2)) / (j! * (-2)^j))}
seq(n)={my(v=vector(n, n, b(n)*binomial(2*n, n)), u=vector(n)); for(n=1, #u, u[n]=v[n] - sum(k=3, n-3, binomial(2*n-1,2*k)*v[k]*u[n-k])); u[3..n]/2} \\ Andrew Howroyd, May 22 2018

a(n) = binomial(2*n-1, n)*A001501(n) - Sum_{k=1..n-1} binomial(2*n-1, 2*k) * binomial(2*k, k) * A001501(k) * a(n-k). - Andrew Howroyd, May 22 2018

a(n) ~ 3^(n + 1/2) * n^(3*n) / (sqrt(2) * exp(3*n + 2)). - Vaclav Kotesovec, Feb 17 2024

a(7)-a(8) corrected and a(9)-a(12) computed with nauty by Sean A. Irvine, Jun 27 2017

Terms a(13) and beyond from Andrew Howroyd, May 22 2018

A386204 Number of distinct values of the determinant of an n X n (0,1)-matrix with exactly three 1's in each row and each column.

Original entry on oeis.org

1, 2, 3, 3, 7, 11, 7

Offset: 3

Robert P. P. McKone, Jul 15 2025

a(10) >= 25.

Robert P. P. McKone, Python code to find A386204(n) unique determinants and one example matrix
Robert P. P. McKone, One example matrix for each known determinant for a(3)-a(9)

Cf. A185178 (distinct permanents).

Cf. A001501 (number of n X n (0,1)-matrix with exactly three 1's in each row and each column).

A246602 Number of connected bicubical graphs on 2n labeled nodes of two colors.

Original entry on oeis.org

1, 24, 2040, 297000, 68922000, 24038380800

Offset: 3

N. J. A. Sloane, Sep 08 2014

Should a(7) be 68909400? - Andrew Howroyd, May 23 2018

R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.

R. C. Read, Letter to N. J. A. Sloane, Feb 04 1971

Cf. A001501.

cp's OEIS Frontend

A006714 Number of trivalent bipartite labeled graphs with 2n labeled nodes.

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A079815 Number of equivalent classes of n X n 0-1 matrices with 3 1's in each row and column.

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A225623 Number of ways to arrange 2n queens on an n X n chessboard, with no more than 2 queens in each row, column or diagonal.

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A246599 Number of connected trivalent bipartite labeled graphs with 2n labeled nodes.

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Mathematica

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Extensions

A386204 Number of distinct values of the determinant of an n X n (0,1)-matrix with exactly three 1's in each row and each column.

Views

Author

Keywords

Comments

Links

Crossrefs

A246602 Number of connected bicubical graphs on 2n labeled nodes of two colors.

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Comments

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Crossrefs