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.

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

Author

Benoit Jubin, May 01 2008

Keywords

• nonn
• tabl

Comments

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

Examples

			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.
		

Links

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

Crossrefs

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

Programs

PARI

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

Formula

T(n,1) = 1.

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

Extensions

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