A137975 -id:A137975 - cp's OEIS frontend

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

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

A129620 Square array read by falling antidiagonals: T(n,k) is the number of connected directed multigraphs with loops with n arcs and at most k vertices.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 5, 1, 0, 1, 2, 8, 9, 1, 0, 1, 2, 8, 24, 17, 1, 0, 1, 2, 8, 32, 74, 26, 1, 0, 1, 2, 8, 32, 140, 189, 41, 1, 0, 1, 2, 8, 32, 167, 542, 460, 57, 1, 0, 1, 2, 8, 32, 167, 837, 1964, 989, 81, 1, 0, 1, 2, 8, 32, 167, 928, 4167, 6291, 2021, 106, 1, 0

Offset: 0

Benoit Jubin, May 06 2008

Partial sums of the rows of A139621, i.e., T(n,k) = sum(A139621(n,p),p=0..k).

			1  1  1  1  1  1  ...
0  1  2  2  2  2  ...
0  1  5  8  8  8  ...
0  1  9 24 32 32  ...
0  1  17  (...)
(...)

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Cf. A138107, A139621.

T(n,2) = A138107(n,2) - floor(n/2).

If k >= n+1, T(n,k) = A137975(n).

Name edited by M. F. Hasler, Jul 31 2017

Terms a(32) and beyond from Andrew Howroyd, Oct 22 2019

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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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A129620 Square array read by falling antidiagonals: T(n,k) is the number of connected directed multigraphs with loops with n arcs and at most k vertices.

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