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 1-10 of 10 results.

A035512 Number of unlabeled strongly connected digraphs with n nodes.

Original entry on oeis.org

1, 1, 1, 5, 83, 5048, 1047008, 705422362, 1580348371788, 12139024825260556, 328160951349343885604, 31831080872412589394328804, 11234274997368899732057135454531, 14576252633139820879894296847900227082
Offset: 0

Views

Author

Ronald C. Read

Keywords

References

  • F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 218.
  • V. A. Liskovets, A contribution to the enumeration of strongly connected digraphs, Dokl. AN BSSR, 17 (1973), 1077-1080, MR49#4849.
  • R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
  • R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.

Links

Crossrefs

The labeled version is A003030.
Row sums of A057276.
Column sums of A350753.
Cf. A003085, A049387, A350489, A350752.

Programs

Extensions

a(12) and a(13) added by N. J. A. Sloane from the Robinson report, Oct 17 2006

A057276 Triangle T(n,k) of number of strongly connected digraphs on n unlabeled nodes and with k arcs, k=0..n*(n-1).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 2, 1, 1, 0, 0, 0, 0, 1, 4, 16, 22, 22, 11, 5, 1, 1, 0, 0, 0, 0, 0, 1, 7, 58, 240, 565, 928, 1065, 953, 640, 359, 150, 59, 16, 5, 1, 1, 0, 0, 0, 0, 0, 0, 1, 10, 165, 1281, 6063, 19591, 47049, 87690, 131927, 163632, 170720, 151238, 115122, 75357, 42745, 20891, 8877, 3224, 1039, 286, 76, 17, 5, 1, 1
Offset: 1

Views

Author

Vladeta Jovovic, Goran Kilibarda, Sep 14 2000

Keywords

Examples

			Triangle begins:
  [1] 1;
  [2] 0,0,1;
  [3] 0,0,0,1,2,1,1;
  [4] 0,0,0,0,1,4,16,22,22,11,5,1,1;
  ...
The number of strongly connected digraphs on 3 unlabeled nodes is 5 = 1+2+1+1.

Links

Crossrefs

Row sums give A035512.
Column sums give A350752.
The labeled version is A057273.
Cf. A054733, A057270, A057277, A057278, A057279, A350489.

Programs

Extensions

Terms a(46) and beyond from Andrew Howroyd, Jan 13 2022

A350752 Number of unlabeled strongly connected digraphs with n arcs.

Original entry on oeis.org

1, 0, 1, 1, 3, 6, 25, 91, 442, 2241, 12591, 75180, 478648, 3211245, 22635956, 166828221, 1281518573, 10229858290, 84652925554, 724601312400, 6403522811765, 58327076550161, 546764617643250, 5267719312771122, 52096218005705959, 528285485054771639
Offset: 0

Views

Author

Andrew Howroyd, Jan 13 2022

Keywords

Crossrefs

Row sums of A350753.
Column sums of A057276.
Cf. A035512, A350489, A350751.

Programs

A350750 Triangle read by rows: T(n,k) is the number of strongly connected oriented graphs on n unlabeled nodes with k arcs, n >= 1, k = 0..n*(n-1)/2.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 5, 18, 27, 19, 6, 0, 0, 0, 0, 0, 0, 1, 8, 80, 333, 765, 1122, 1049, 622, 217, 35, 0, 0, 0, 0, 0, 0, 0, 1, 12, 221, 1875, 8971, 28449, 63845, 105556, 130935, 122607, 85926, 43868, 15506, 3403, 353
Offset: 1

Views

Author

Andrew Howroyd, Jan 13 2022

Keywords

Examples

			Triangle begins:
  [1] 1;
  [2] 0, 0;
  [3] 0, 0, 0, 1;
  [4] 0, 0, 0, 0, 1, 2, 1;
  [5] 0, 0, 0, 0, 0, 1, 5, 18, 27,  19,   6;
  [6] 0, 0, 0, 0, 0, 0, 1,  8, 80, 333, 765, 1122, 1049, 622, 217, 35;
  ...

Links

Crossrefs

Row sums are A350489.
Column sums are A350751.
The labeled version is A350731.
Cf. A057276 (digraphs), A350733, A350734.

Programs

  • PARI
    \\ See PARI link in A350489 for program code.
{ my(A=A350750rows(7)); for(n=1, #A, print(A[n])) }

A139622 Triangle read by rows: T(n,k) is the number of strongly connected directed multigraphs with loops, with n arcs and k vertices.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 4, 1, 1, 10, 19, 6, 1, 1, 19, 73, 59, 9, 1, 1, 28, 208, 350, 138, 12, 1, 1, 44, 534, 1670, 1361, 301, 16, 1, 1, 60, 1215, 6476, 9724, 4364, 575, 20, 1, 1, 85, 2542, 21898, 55707, 45284, 12131, 1042, 25, 1, 1, 110, 4951, 65789, 268329, 365063, 175416, 30090, 1749, 30, 1
Offset: 1

Views

Author

Benoit Jubin, May 01 2008

Keywords

Examples

			Triangle begins:
    1
    1    1
    1    2    1
    1    6    4    1
    1   10   19    6    1
    1   19   73   59    9    1
    1   28  208  350  138   12    1
    1   44  534 1670 1361  301   16  1
    ...
T(4 edges, 2 vertices)=6: one graph 1->1, 1->1, 2->1, 1->2; one graph 1->1, 2->1, 2->1, 1->2; one graph 1->1, 1->2, 1->2, 2->1; one graph 1->1, 1->2, 2->1, 2->2; one graph 2->1, 2->1, 2->1, 1->2; one graph 1->2, 1->2, 2->1, 2->1.
T(4 edges, 3 vertices)=4: one graph 1->1, 2->1, 3->2, 1->3; one graph 2->1, 2->1, 3->2, 1->3; one graph 2->1, 3->1, 1->2, 1->3; one graph 2->1, 3->1, 1->2, 2->3.

Links

Crossrefs

Row sums are A139627.
Cf. A136564, A139621, A143841, A350489, A350753.

Programs

Formula

T(n,1) = T(n,n) = 1.
T(n,2) = A139621(n,2) - n(n+1)/2.

Extensions

More terms from R. J. Mathar, Aug 11 2017
Terms a(34) and beyond from Andrew Howroyd, Jan 14 2022

A139627 Number of strongly connected directed multigraphs with loops allowed and with n arcs.

Original entry on oeis.org

1, 1, 2, 4, 12, 37, 162, 738, 3928, 22436, 138716, 911529, 6339770, 46336941, 354453138, 2826472249, 23423053967, 201179882629, 1786791372857, 16377359709120, 154644691266520, 1502016160624186, 14985219655673207, 153377735526218010, 1608741204839374373
Offset: 0

Views

Author

Benoit Jubin, May 01 2008

Keywords

Comments

The term a(0)=1 can be interpreted as either a singleton vertex or the graph with no vertices. - Andrew Howroyd, Jan 14 2022

Links

Crossrefs

Row sums of A139622.
Cf. A052171, A350489, A350752.

Programs

Extensions

3 more terms from R. J. Mathar, Aug 04 2017
Terms a(7) and beyond from Andrew Howroyd, Jan 14 2022

A350730 Number of strongly connected oriented graphs on n labeled nodes.

Original entry on oeis.org

1, 0, 2, 66, 7998, 2895570, 3015624078, 8890966977354, 74079608267459142, 1754419666770364130730, 119163820122708911990211222, 23431180614718394105521543222866, 13448652672256961901980839022683943838, 22684139279519345808802725789494254587951810
Offset: 1

Views

Author

Andrew Howroyd, Jan 11 2022

Keywords

Links

Crossrefs

The unlabeled version is A350489.
Row sums of A350731.
Cf. A003030, A054941.

Programs

  • PARI
    StrongO(14) \\ See A350731 for program code.

A350753 Triangle read by rows: T(n,k) is the number of unlabeled strongly connected digraphs with n arcs and k vertices, n >= 0, k = 1..n+1.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 4, 1, 0, 0, 0, 1, 16, 7, 1, 0, 0, 0, 0, 22, 58, 10, 1, 0, 0, 0, 0, 22, 240, 165, 14, 1, 0, 0, 0, 0, 11, 565, 1281, 365, 18, 1, 0, 0, 0, 0, 5, 928, 6063, 4838, 733, 23, 1, 0, 0, 0, 0, 1, 1065, 19591, 38516, 14661, 1317, 28, 1, 0
Offset: 0

Views

Author

Andrew Howroyd, Jan 14 2022

Keywords

Examples

			Triangle begins:
  1;
  0, 0;
  0, 1, 0;
  0, 0, 1,  0;
  0, 0, 2,  1,   0;
  0, 0, 1,  4,   1,    0;
  0, 0, 1, 16,   7,    1,   0;
  0, 0, 0, 22,  58,   10,   1,  0;
  0, 0, 0, 22, 240,  165,  14,  1, 0;
  0, 0, 0, 11, 565, 1281, 365, 18, 1, 0;

Crossrefs

Row sums are A350752.
Column sums are A035512.
Cf. A057276 (transpose), A350450, A350489.

Programs

  • PARI
    \\ See PARI link in A350489 for program code.
my(A=A350753rows(10)); for(n=1, #A, print(A[n]))

A143841 Table read by antidiagonals: T(n,k) is the number of strongly connected directed multigraphs with loops with n arcs and up to k vertices.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 4, 7, 1, 0, 1, 1, 2, 4, 11, 11, 1, 0, 1, 1, 2, 4, 12, 30, 20, 1, 0, 1, 1, 2, 4, 12, 36, 93, 29, 1, 0, 1, 1, 2, 4, 12, 37, 152, 237, 45, 1, 0, 1, 1, 2, 4, 12, 37, 161, 587, 579, 61, 1, 0
Offset: 0

Views

Author

Benoit Jubin, Sep 02 2008

Keywords

Examples

			Array begins:
=============================================
n\k | 0 1  2   3    4    5    6    7    8
----+----------------------------------------
  0 | 1 1  1   1    1    1    1    1    1 ...
  1 | 0 1  1   1    1    1    1    1    1 ...
  2 | 0 1  2   2    2    2    2    2    2 ...
  3 | 0 1  3   4    4    4    4    4    4 ...
  4 | 0 1  7  11   12   12   12   12   12 ...
  5 | 0 1 11  30   36   37   37   37   37 ...
  6 | 0 1 20  93  152  161  162  162  162 ...
  7 | 0 1 29 237  587  725  737  738  738 ...
  8 | 0 1 45 579 2249 3610 3911 3927 3928 ...
  ...

Links

Crossrefs

Partial sums of the rows of A139622.
Main diagonal is A139627.
Cf. A138107, A129620, A143842, A350489.

Programs

  • PARI
    \\ See PARI link in A350489 for program code.
A(n)={my(data=A139622rows(n), M=matrix(n+1, n+1, i, j, if(i==1, 1, sum(k=1, min(i-1,j-1), data[i-1][k])))); M}
{ my(M=A(8)); for(n=1, #M~, print(M[n,])) } \\ Andrew Howroyd, Jan 14 2022

Formula

T(n,k) = Sum_{p=0..k} A139622(n,p).
T(n,k) = A139627(n) for k >= n.
T(n,2) = A129620(n,2) - n*(n-1)/2.

Extensions

Name clarified and terms a(32) and beyond from Andrew Howroyd, Jan 14 2022

A350751 Number of unlabeled strongly connected oriented graphs with n arcs.

Original entry on oeis.org

1, 0, 0, 1, 1, 3, 7, 27, 120, 590, 3172, 18291, 112597, 731646, 5002261, 35805035, 267427870, 2077890719, 16753007782, 139843832074, 1206213748404, 10731971520412, 98340872850392, 926788397662602, 8971598464914888, 89105060389927172, 907030917078092542
Offset: 0

Views

Author

Andrew Howroyd, Jan 14 2022

Keywords

Crossrefs

Column sums of A350750.
Cf. A350489, A350752.

Programs

Showing 1-10 of 10 results.

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

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