A139621 - OEIS

%I #44 Mar 23 2023 23:08:53

%S 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,

%T 2203,1407,350,1,56,932,5302,12382,13372,6790,1376,1,80,1940,16549,

%U 58237,96456,80736,33628,5743,1,105,3743,46566,237904,573963,717114,482730,168645,24635

%N 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.

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

%H Andrew Howroyd, <a href="/A139621/b139621.txt">Table of n, a(n) for n = 0..1325</a>

%H R. J. Mathar, <a href="http://arxiv.org/abs/1709.09000">Statistics on Small Graphs</a>, arXiv:1709.09000 (2017) Table 71.

%F T(n,1) = 1.

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

%e Triangle begins:

%e      1

%e      1     1

%e      1     4     3

%e      1     8    15     8

%e      1    16    57    66    27

%e      1    25   163   353   295    91

%e      1    40   419  1504  2203  1407   350

%e      1    56   932  5302 12382 13372  6790  1376

%e 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.

%e T(2 arcs, 3 vertices) = 3: one graph 2->1, 3->1; one graph 2->1, 3->2; one graph 2->1, 2->3.

%o (PARI)

%o 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))}

%o 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}

%o 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])}

%o G(n, x)={my(s=0); forpart(p=n, s+=permcount(p)/edges(p,i->1-x^i)); s/n!}

%o T(n)={Mat([Col(p+O(y^n), -n) | p<-InvEulerMT(vector(n, k, G(k, y + O(y^n))))])}

%o {my(A=T(10)); for(n=1, #A, print(A[n,1..n]))} \\ _Andrew Howroyd_, Oct 22 2019

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

%K nonn,tabl

%O 0,5

%A _Benoit Jubin_, May 01 2008

%E Prepended a(0)=1 to have a regular triangle, _Joerg Arndt_, Apr 14 2013

%E More terms from _R. J. Mathar_, Jul 31 2017

%E Terms a(34) and beyond from _Andrew Howroyd_, Oct 22 2019