A136564 -id:A136564 - cp's OEIS frontend

A144047 Limiting values of A136564.

1, 4, 22, 111, 570, 2917, 15112, 78900, 416525, 2221529, 11974498, 65211746, 358742016, 1992975905, 11178149883, 63279615534, 361464920864, 2082869242399, 12104344580384, 70924864962233, 418922049779244

Offset: 0

Benoit Jubin and Vladeta Jovovic, Sep 08 2008

Cf. A136564, A138107.

\\ Needs G function from A138107.
a(n)={if(n==0, 1, my(m=2*n); polcoef(G(3*n, x + O(x*x^m)) - G(3*n-1, x + O(x*x^m)), m))} \\ Andrew Howroyd, Apr 20 2020

a(n) = A136564(2*n, 3*n) for n > 0. - Andrew Howroyd, Apr 20 2020

Offset corrected and a(7)-a(20) from Andrew Howroyd, Apr 20 2020

A138107 Infinite square array: T(n,k) = number of directed multigraphs with loops with n arcs and k vertices; read by falling antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 6, 1, 0, 1, 2, 10, 10, 1, 0, 1, 2, 11, 31, 19, 1, 0, 1, 2, 11, 47, 90, 28, 1, 0, 1, 2, 11, 51, 198, 222, 44, 1, 0, 1, 2, 11, 52, 269, 713, 520, 60, 1, 0, 1, 2, 11, 52, 291, 1270, 2423, 1090, 85, 1, 0, 1, 2, 11, 52, 295, 1596, 5776, 7388, 2180, 110, 1, 0

Offset: 0

Benoit Jubin, May 03 2008

Partial sums of the rows of A136564.

			The array begins:
   1, 1,   1,    1,     1,     1,     1,     1,     1, ...
   0, 1,   2,    2,     2,     2,     2,     2,     2, ...
   0, 1,   6,   10,    11,    11,    11,    11,    11, ...
   0, 1,  10,   31,    47,    51,    52,    52,    52, ...
   0, 1,  19,   90,   198,   269,   291,   295,   296,  296, ...
   0, 1,  28,  222,   713,  1270,  1596,  1697,  1719, 1723, ...
   0, 1,  44,  520,  2423,  5776,  8838, 10425, 10922, ...
   0, 1,  60, 1090,  7388, 24032, 46384, ...
   0, 1,  85, 2180, 21003, 93067, ...
   0, 1, 110, 4090, ...
   ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO] (2017) Table 79.

Columns k=0..4 are: A000007, A000012, A005993, A050927, A050929.
Main diagonal is A362387.
Cf. A052171, A136564, A333361.

permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
edges(v,t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i],v[j])); t(v[i]*v[j]/g)^(2*g))) * prod(i=1, #v, t(v[i])^v[i])}
G(n, x)={my(s=0); forpart(p=n, s+=permcount(p)/edges(p,i->1-x^i)); s/n!}
T(n)={Mat(vector(n+1, k, Col(O(y*y^n) + G(k-1, y + O(y*y^n)))))}
{my(A=T(10)); for(n=1, #A, print(A[n,]))} \\ Andrew Howroyd, Oct 22 2019

T(n,k) = Sum_{p=0..k} A136564(n,p).

If k >= 2n, T(n,k) = A052171(n).

More terms from Vladeta Jovovic and Benoit Jubin, Sep 10 2008

A052171 Number of directed multigraphs with loops on an infinite set of nodes containing a total of n arcs.

Original entry on oeis.org

1, 2, 11, 52, 296, 1724, 11060, 74527, 533046, 3999187, 31412182, 257150093, 2188063401, 19299062896, 176059781439, 1657961491087, 16089088019098, 160643776819423, 1648068916722737, 17351137043998280, 187255329043638437, 2069426416836401375, 23397468305569068113, 270406562951254606048, 3191908298072118225550, 38454691427657997701136

Offset: 0

Vladeta Jovovic, Jan 26 2000

nonn

Row sums of A136564, limiting values of A138107. - Benoit Jubin, May 13 2008

Euler transform of A137975. - M. F. Hasler, Jul 31 2017

Andrew Howroyd, Table of n, a(n) for n = 0..50
Banglei Guan, Ji Zhao, and Laurent Kneip, Six-Point Method for Multi-Camera Systems with Reduced Solution Space, arXiv:2402.18066 [cs.CV], 2024. See p. 10.
J.-C. Novelli, J.-Y. Thibon and N. M. Thiery, Algèbres de Hopf de graphes, C.R. Acad. Sci. Paris (Comptes Rendus Mathématique), 339 (2004), 607-610.
Sanjaye Ramgoolam, Permutation Invariant Gaussian Matrix Models, arXiv:1809.07559 [hep-th], 2018.

Cf. A050535, A007717, A005993, A050927, A050929.

Cf. A104209. Cf. A137975 (connected).

a(n) = A138107(2*n,n). - Max Alekseyev, Oct 17 2017

a(16)-a(25) from Max Alekseyev, Jun 21 2011

A139621 Triangle read by rows: T(n,k) is the number of connected directed multigraphs with loops and no vertex of degree 0, with n arcs and k vertices.

Original entry on oeis.org

1, 1, 1, 1, 4, 3, 1, 8, 15, 8, 1, 16, 57, 66, 27, 1, 25, 163, 353, 295, 91, 1, 40, 419, 1504, 2203, 1407, 350, 1, 56, 932, 5302, 12382, 13372, 6790, 1376, 1, 80, 1940, 16549, 58237, 96456, 80736, 33628, 5743, 1, 105, 3743, 46566, 237904, 573963, 717114, 482730, 168645, 24635

Offset: 0

Benoit Jubin, May 01 2008

Length of the n-th row: n+1.

			Triangle begins:
     1
     1     1
     1     4     3
     1     8    15     8
     1    16    57    66    27
     1    25   163   353   295    91
     1    40   419  1504  2203  1407   350
     1    56   932  5302 12382 13372  6790  1376
T(2 arcs, 2 vertices) = 4: one graph 1->1, 2->1; one graph with 1->1, 1->2; one graph with 2->1, 2->1, one graph with 1->2, 2->1.
T(2 arcs, 3 vertices) = 3: one graph 2->1, 3->1; one graph 2->1, 3->2; one graph 2->1, 2->3.

Andrew Howroyd, Table of n, a(n) for n = 0..1325
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 (2017) Table 71.

Cf. A129620, A136564, A139622, A137975 (row sums), A000238 (diagonal).

InvEulerMT(u)={my(n=#u, p=log(1+x*Ser(u)), vars=variables(p)); Vec(serchop( sum(i=1, n, moebius(i)*substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i), 1))}
permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
edges(v,t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i],v[j])); t(v[i]*v[j]/g)^(2*g))) * prod(i=1, #v, t(v[i])^v[i])}
G(n, x)={my(s=0); forpart(p=n, s+=permcount(p)/edges(p,i->1-x^i)); s/n!}
T(n)={Mat([Col(p+O(y^n), -n) | p<-InvEulerMT(vector(n, k, G(k, y + O(y^n))))])}
{my(A=T(10)); for(n=1, #A, print(A[n,1..n]))} \\ Andrew Howroyd, Oct 22 2019

T(n,1) = 1.

T(n,2) = A136564(n,2) - floor(n/2).

Prepended a(0)=1 to have a regular triangle, Joerg Arndt, Apr 14 2013
More terms from R. J. Mathar, Jul 31 2017
Terms a(34) and beyond from Andrew Howroyd, Oct 22 2019

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

Benoit Jubin, May 01 2008

			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.

Andrew Howroyd, Table of n, a(n) for n = 1..820 (rows 1..40)
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 (2017) Table 73.

Row sums are A139627.

Cf. A136564, A139621, A143841, A350489, A350753.

\\ See PARI link in A350489 for program code.
{ my(A=A139622rows(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 14 2022

T(n,1) = T(n,n) = 1.

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

More terms from R. J. Mathar, Aug 11 2017

Terms a(34) and beyond from Andrew Howroyd, Jan 14 2022

A139623 Table read by rows: T(n,k) is the number of directed multigraphs with loops and no vertex of degree 0, with n arcs and k vertices, which are transitive (the existence of a path between two points implies the existence of an arc between those two points).

Original entry on oeis.org

1, 1, 1, 4, 3, 1, 1, 7, 12, 9, 3, 1, 1, 13

Offset: 1

Benoit Jubin, May 01 2008

Length of the n^th row: 2n.
T(n,1) = T(n,2n) = 1 and T(n,2n-1) = 3 if n>1.
Row sums: A139628.

			Triangle begins
1, 1;
1, 4, 3, 1;
1, 7, 12, 9, 3, 1;

Cf. A136564, A139624, A139625.

A327615 Irregular triangle read by rows: T(n,k) is the number of unlabeled multigraphs with loops allowed and n edges covering k vertices, n >= 1, 1 <= k <= 2*n.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 1, 5, 8, 6, 2, 1, 1, 8, 19, 25, 16, 7, 2, 1, 1, 11, 40, 73, 73, 47, 19, 7, 2, 1, 1, 15, 77, 194, 263, 232, 133, 58, 20, 7, 2, 1, 1, 19, 132, 454, 835, 951, 719, 397, 164, 61, 20, 7, 2, 1, 1, 24, 217, 984, 2385, 3507, 3365, 2306, 1177, 490, 175, 62, 20, 7, 2, 1

Offset: 1

Andrew Howroyd, Oct 23 2019

Covering k vertices means there are no vertices of degree zero.

			Triangle begins:
  1,  1;
  1,  3,   2,   1;
  1,  5,   8,   6,   2,   1;
  1,  8,  19,  25,  16,   7,   2,   1;
  1, 11,  40,  73,  73,  47,  19,   7,   2,  1;
  1, 15,  77, 194, 263, 232, 133,  58,  20,  7,  2, 1;
  1, 19, 132, 454, 835, 951, 719, 397, 164, 61, 20, 7, 2, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..650 (rows 1..25)

Row sums are A007717.
Columns k=2..3 are A024206, A327728.
Cf. A136564, A290428, A309936, A322115.

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c+1)\2)*if(c%2, 1, t(c/2)))}
C(n,m)={my(s=O(x*x^m)); forpart(p=n, s+=permcount(p)/edges(p, i->1-x^i+O(x*x^m))); Col(s/n!)}
T(m) = {my(n=2*m, A=Mat(vector(n+1, n, C(n-1,m)))); A[2..m+1, 2..n+1]-A[2..m+1, 1..n]}
{ my(A=T(8)); for(n=1, matsize(A)[1], print(A[n, 1..2*n])) }

T(n,k) = A290428(n,k) - A290428(n,k-1).

cp's OEIS Frontend

A144047 Limiting values of A136564.

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A138107 Infinite square array: T(n,k) = number of directed multigraphs with loops with n arcs and k vertices; read by falling antidiagonals.

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A052171 Number of directed multigraphs with loops on an infinite set of nodes containing a total of n arcs.

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A139621 Triangle read by rows: T(n,k) is the number of connected directed multigraphs with loops and no vertex of degree 0, with n arcs and k vertices.

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

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A139623 Table read by rows: T(n,k) is the number of directed multigraphs with loops and no vertex of degree 0, with n arcs and k vertices, which are transitive (the existence of a path between two points implies the existence of an arc between those two points).

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A327615 Irregular triangle read by rows: T(n,k) is the number of unlabeled multigraphs with loops allowed and n edges covering k vertices, n >= 1, 1 <= k <= 2*n.

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