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 1-6 of 6 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

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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

A350449 Triangle read by rows: T(n,k) is the number of weakly connected acyclic digraphs on n unlabeled nodes with k arcs, n >= 1, k = 0..(n-1)*n/2.

Original entry on oeis.org

1, 0, 1, 0, 0, 3, 1, 0, 0, 0, 8, 9, 6, 1, 0, 0, 0, 0, 27, 54, 79, 63, 33, 10, 1, 0, 0, 0, 0, 0, 91, 320, 732, 1136, 1281, 1056, 649, 281, 85, 15, 1, 0, 0, 0, 0, 0, 0, 350, 1788, 6012, 14378, 26529, 38407, 44621, 41638, 31321, 18843, 8983, 3325, 920, 180, 21, 1
Offset: 1

Views

Author

Andrew Howroyd, Dec 31 2021

Keywords

Examples

			Triangle begins:
  [1] 1;
  [2] 0, 1;
  [3] 0, 0, 3, 1;
  [4] 0, 0, 0, 8,  9,  6,  1;
  [5] 0, 0, 0, 0, 27, 54, 79, 63, 33, 10, 1;
  ...

Links

Crossrefs

Row sums are A101228.
Columns sums are A350451.
Leading diagonal is A000238.
Cf. A350447 (not necessarily connected), A350450 (transpose).

Programs

  • PARI
    \\ See PARI link in A122078 for program code.
{ my(T=WeakAcyclicDigraphsByArcs(6)); 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.

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])) }

A350789 Triangle read by rows: T(n,k) is the number of unlabeled weakly connected digraphs with n arcs and k vertices, k = 1..n+1.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 0, 4, 8, 0, 0, 4, 22, 27, 0, 0, 1, 37, 108, 91, 0, 0, 1, 47, 326, 582, 350, 0, 0, 0, 38, 667, 2432, 3024, 1376, 0, 0, 0, 27, 1127, 7694, 17314, 16008, 5743, 0, 0, 0, 13, 1477, 19646, 74676, 117312, 84494, 24635
Offset: 0

Views

Author

Andrew Howroyd, Jan 28 2022

Keywords

Examples

			Triangle begins:
  1;
  0, 1;
  0, 1, 3;
  0, 0, 4,  8;
  0, 0, 4, 22,   27;
  0, 0, 1, 37,  108,   91;
  0, 0, 1, 47,  326,  582,   350;
  0, 0, 0, 38,  667, 2432,  3024,  1376;
  0, 0, 0, 27, 1127, 7694, 17314, 16008, 5743;
  ...

Links

Crossrefs

Row sums are A053454.
Column sums are A003085.
Main diagonal is A000238.
Cf. A054733 (transpose), A350450 (acyclic), A350753 (strongly connected).

Programs

  • PARI
    \\ See A054733 for G, InvEulerMTS
T(n)={my(p=InvEulerMTS(sum(i=0, n, G(i, y+O(y^n))*x^i, O(x*x^n)))); vector(n, n, Vec(O(x^n)+polcoef(p,n-1,y)/x, -n))}
{ my(A=T(10)); for(n=1, #A, print(A[n])) }

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]))
Showing 1-6 of 6 results.

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

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