%I #9 Feb 07 2024 23:27:39

%S 0,0,0,0,0,1,0,0,0,1,0,0,0,1,1,0,0,0,1,3,2,0,0,0,0,3,5,2,0,0,0,0,2,11,

%T 9,3,0,0,0,0,1,15,32,16,4,0,0,0,0,1,12,63,76,25,5,0,0,0,0,0,8,89,234,

%U 162,39,6,0,0,0,0,0,5,97,515,730,332,60,9

%N Triangle read by rows: T(n,k) is the number of unlabeled simple graphs with n edges and k vertices and without endpoints or isolated vertices.

%H Andrew Howroyd, <a href="/A369932/b369932.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)

%F T(n,k) = A123551(k,n) - A123551(k-1,n).

%e Triangle begins:

%e 0;

%e 0, 0;

%e 0, 0, 1;

%e 0, 0, 0, 1;

%e 0, 0, 0, 1, 1;

%e 0, 0, 0, 1, 3, 2;

%e 0, 0, 0, 0, 3, 5, 2;

%e 0, 0, 0, 0, 2, 11, 9, 3;

%e 0, 0, 0, 0, 1, 15, 32, 16, 4;

%e 0, 0, 0, 0, 1, 12, 63, 76, 25, 5;

%e ...

%o (PARI)

%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)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))}

%o G(n) = {my(s=O(x*x^n)); sum(k=0, n, forpart(p=k, s+=permcount(p) * edges(p, w->1+y^w+O(y*y^n)) * x^k * prod(i=1, #p, 1-(y*x)^p[i], 1+O(x^(n-k+1))) / k!)); s*(1-x)}

%o T(n)={my(r=Vec(substvec(G(n),[x,y],[y,x]))); vector(#r-1, i, Vecrev(Pol(r[i+1]/y),i)) }

%o { my(A=T(12)); for(i=1, #A, print(A[i])) }

%Y Row sums are A369290.

%Y Column sums are A261919.

%Y Main diagonal is A008483.

%Y Cf. A342557 (connected), A123551 (without endpoints).

%K nonn,tabl

%O 1,20

%A _Andrew Howroyd_, Feb 07 2024