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 1-8 of 8 results.

A122078 Triangle read by rows: T(n,k) is the number of unlabeled acyclic digraphs with n >= 0 nodes and n-k outnodes (0 <= k <= n).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 11, 16, 0, 1, 4, 25, 108, 164, 0, 1, 5, 47, 422, 2168, 3341, 0, 1, 6, 78, 1251, 15484, 88747, 138101, 0, 1, 7, 120, 3124, 79836, 1215783, 7409117, 11578037, 0, 1, 8, 174, 6925, 333004, 11620961, 199203464, 1252610909, 1961162564, 0
Offset: 0

Views

Author

N. J. A. Sloane, Oct 18 2006

Keywords

Examples

			Triangle T(n,k) begins:
  1:
  1, 0;
  1, 1,  0;
  1, 2,  3,    0;
  1, 3, 11,   16,     0;
  1, 4, 25,  108,   164,     0;
  1, 5, 47,  422,  2168,  3341,      0;
  1, 6, 78, 1251, 15484, 88747, 138101, 0;
  ...

References

  • R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.

Links

Crossrefs

Row sums give A003087.
Diagonals include A000007, A350415.
Cf. A058876 (labeled case), A350447, A350448, A350449, A350450.

Programs

  • PARI
    \\ See link for program code.
{ my(T=AcyclicDigraphsByNonSources(8)); for(n=1, #T, print(T[n])) } \\ Andrew Howroyd, Dec 31 2021

Extensions

Zero terms inserted by Andrew Howroyd, Dec 29 2021

A350450 Triangle read by rows: T(n,k) is the number of unlabeled weakly connected acyclic digraphs with n arcs and k vertices, n >= 0, k = 1..n+1.

Original entry on oeis.org

1, 0, 1, 0, 0, 3, 0, 0, 1, 8, 0, 0, 0, 9, 27, 0, 0, 0, 6, 54, 91, 0, 0, 0, 1, 79, 320, 350, 0, 0, 0, 0, 63, 732, 1788, 1376, 0, 0, 0, 0, 33, 1136, 6012, 9933, 5743, 0, 0, 0, 0, 10, 1281, 14378, 45225, 54502, 24635, 0, 0, 0, 0, 1, 1056, 26529, 151848, 322736, 298250, 108968
Offset: 0

Views

Author

Andrew Howroyd, Dec 31 2021

Keywords

Examples

			Triangle begins:
  1;
  0, 1;
  0, 0, 3;
  0, 0, 1, 8;
  0, 0, 0, 9, 27;
  0, 0, 0, 6, 54,   91;
  0, 0, 0, 1, 79,  320,  350;
  0, 0, 0, 0, 63,  732, 1788, 1376;
  0, 0, 0, 0, 33, 1136, 6012, 9933, 5743;
  ...

Links

Crossrefs

Main diagonal is A000238.
Row sums are A350451.
Column sums are A101228.
Cf. A122078, A350449 (transpose).

Programs

  • PARI
    \\ See PARI link in A122078 for program code.
{ my(T=WeakAcyclicDigraphsTr(10)); for(n=1, #T, print(T[n])); }

A101228 Number of weakly connected acyclic digraphs on n unlabeled nodes.

Original entry on oeis.org

1, 1, 4, 24, 267, 5647, 237317, 20035307, 3404385285, 1162502511721, 796392234736238, 1093228137893084112, 3004752537725051647790, 16527844667281561960220731, 181891583847006693859132403681, 4004313473818592854334088690859030
Offset: 1

Views

Author

Vladeta Jovovic, Jan 22 2005

Keywords

Comments

The multiset transformation gives the number acyclic digraphs on n unlabeled nodes with k components:
1 ;
1 , 1 ;
4 , 1 , 1 ;
24 , 5 , 1 , 1 ;
267 , 28 , 5 , 1 , 1 ;
5647 , 301 , 29 , 5 , 1 , 1 ;
237317 , 6010 , 305 , 29 , 5 , 1 , 1 ; R. J. Mathar, Mar 21 2019

Links

Crossrefs

Row sums of A350449.
Column sums of A350450.
Cf. A003087 (Euler trans.), A082402, A350451.

A350447 Triangle read by rows: T(n,k) is the number of acyclic digraphs on n unlabeled nodes with k arcs, n >=0, k = 0..(n-1)*n/2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 4, 9, 9, 6, 1, 1, 1, 4, 12, 37, 60, 80, 63, 33, 10, 1, 1, 1, 4, 13, 51, 163, 407, 796, 1169, 1291, 1057, 649, 281, 85, 15, 1, 1, 1, 4, 13, 54, 215, 846, 2690, 7253, 15703, 27596, 39057, 44902, 41723, 31336, 18844, 8983, 3325, 920, 180, 21, 1
Offset: 0

Views

Author

Andrew Howroyd, Dec 31 2021

Keywords

Examples

			Triangle begins:
  [0] 1;
  [1] 1;
  [2] 1, 1;
  [3] 1, 1, 3,  1;
  [4] 1, 1, 4,  9,  9,  6,  1;
  [5] 1, 1, 4, 12, 37, 60, 80, 63, 33, 10, 1;
  ...

Links

Crossrefs

The labeled version is A081064.
Row sums are A003087.
Cf. A122078, A350449.

Programs

  • PARI
    \\ See PARI link in A122078 for program code.
{ my(T=AcyclicDigraphsByArcs(6)); for(n=1, #T, print(T[n])) }

A350451 Number of weakly connected acyclic digraphs with n arcs.

Original entry on oeis.org

1, 1, 3, 9, 36, 151, 750, 3959, 22857, 140031, 909388, 6202031, 44256875, 328994157, 2540242646, 20317980102, 167980915848, 1432808198569, 12587788263807, 113739153822878, 1055610955120803, 10051265993496814, 98083750658261085, 979961276867802001, 10015362142357613001
Offset: 0

Views

Author

Andrew Howroyd, Dec 31 2021

Keywords

Links

Crossrefs

Row sums of A350450.
Column sums of A350449.
Cf. A101228, A122078.

Programs

  • PARI
    \\ See PARI link in A122078 for program code.
{ my(T=WeakAcyclicDigraphsTr(15)); vector(#T, n, vecsum(T[n])) }

A350488 Triangle read by rows: T(n,k) is the number of acyclic digraphs on n unlabeled nodes with k arcs and a global source, n >= 1, k = 0..n*(n-1)/2.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 1, 0, 0, 0, 4, 6, 5, 1, 0, 0, 0, 0, 9, 25, 47, 46, 27, 9, 1, 0, 0, 0, 0, 0, 20, 95, 297, 582, 783, 738, 501, 235, 75, 14, 1, 0, 0, 0, 0, 0, 0, 48, 337, 1575, 4941, 11295, 19404, 25847, 26966, 22195, 14380, 7280, 2831, 816, 165, 20, 1
Offset: 1

Views

Author

Andrew Howroyd, Jan 01 2022

Keywords

Examples

			Triangle begins:
  [1] 1;
  [2] 0, 1;
  [3] 0, 0, 2, 1;
  [4] 0, 0, 0, 4, 6,  5,  1;
  [5] 0, 0, 0, 0, 9, 25, 47, 46, 27, 9, 1;
  [6] 0, 0, 0, 0, 0, 20, 95, 297, 582, 783, 738, 501, 235, 75, 14, 1;
  ...

Links

Crossrefs

Row sums are A350415.
Column sums are A350490.
Leading diagonal is A000081.
The labeled version is A350487.
Cf. A122078, A350447, A350449, A350491.

Programs

  • PARI
    \\ See PARI link in A122078 for program code.
{ my(A=A350488rows(7)); for(i=1, #A, print(A[i])) }

A350491 Triangle read by rows: T(n,k) is the number of acyclic digraphs on n unlabeled nodes with k arcs and a global source and sink, n >= 1, k = 0..n*(n-1)/2.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 4, 4, 1, 0, 0, 0, 0, 1, 9, 25, 32, 22, 8, 1, 0, 0, 0, 0, 0, 1, 17, 92, 259, 441, 496, 379, 195, 66, 13, 1, 0, 0, 0, 0, 0, 0, 1, 28, 259, 1286, 4026, 8754, 13930, 16686, 15289, 10785, 5842, 2397, 722, 151, 19, 1
Offset: 1

Views

Author

Andrew Howroyd, Jan 08 2022

Keywords

Examples

			Triangle begins:
  [1] 1;
  [2] 0, 1;
  [3] 0, 0, 1, 1;
  [4] 0, 0, 0, 1, 4, 4,  1;
  [5] 0, 0, 0, 0, 1, 9, 25, 32,  22,   8,   1;
  [6] 0, 0, 0, 0, 0, 1, 17, 92, 259, 441, 496, 379, 195, 66, 13, 1;
  ...

Links

Crossrefs

Row sums are A345258.
Column sums are A350492.
Cf. A122078, A350447, A350449, A350488.

Programs

  • PARI
    \\ See PARI link in A122078 for program code.
{ my(A=A350491rows(7)); for(i=1, #A, print(A[i])) }

A350909 Triangle read by rows: T(n,k) is the number of weakly connected acyclic digraphs on n labeled nodes with k arcs, k=0..n*(n-1).

Original entry on oeis.org

1, 0, 2, 0, 0, 12, 6, 0, 0, 0, 128, 186, 108, 24, 0, 0, 0, 0, 2000, 5640, 7840, 6540, 3330, 960, 120, 0, 0, 0, 0, 0, 41472, 189480, 456720, 730830, 832370, 690300, 416160, 178230, 51480, 9000, 720, 0, 0, 0, 0, 0, 0, 1075648, 7178640, 26035800, 65339820
Offset: 1

Views

Author

Andrew Howroyd, Jan 29 2022

Keywords

Examples

			Triangle begins:
  [1] 1;
  [2] 0, 2;
  [3] 0, 0, 12,   6;
  [4] 0, 0,  0, 128,  186,  108,   24;
  [5] 0, 0,  0,   0, 2000, 5640, 7840, 6540, 3330, 960, 120;
  ...

Links

Crossrefs

Row sums are A082402.
Leading diagonal is A097629.
The unlabeled version is A350449.
Cf. A057273, A062735, A081064.

Programs

  • PARI
    G(n)={my(v=vector(n+1)); v[1]=1; for(n=1, n, v[n+1]=sum(k=1, n, -(-1)^k*(1+y)^(k*(n-k))*v[n-k+1]/k!))/n!; Ser(v)}
row(n)={Vecrev(n!*polcoef(log(G(n)), n))}
{ for(n=1, 6, print(row(n))) }
Showing 1-8 of 8 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.