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A369932 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.

%I A369932 #9 Feb 07 2024 23:27:39
%S A369932 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 A369932 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 A369932 162,39,6,0,0,0,0,0,5,97,515,730,332,60,9
%N A369932 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 A369932 Andrew Howroyd, <a href="/A369932/b369932.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%F A369932 T(n,k) = A123551(k,n) - A123551(k-1,n).
%e A369932 Triangle begins:
%e A369932   0;
%e A369932   0, 0;
%e A369932   0, 0, 1;
%e A369932   0, 0, 0, 1;
%e A369932   0, 0, 0, 1, 1;
%e A369932   0, 0, 0, 1, 3,  2;
%e A369932   0, 0, 0, 0, 3,  5,  2;
%e A369932   0, 0, 0, 0, 2, 11,  9,  3;
%e A369932   0, 0, 0, 0, 1, 15, 32, 16,  4;
%e A369932   0, 0, 0, 0, 1, 12, 63, 76, 25, 5;
%e A369932   ...
%o A369932 (PARI)
%o A369932 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 A369932 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 A369932 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 A369932 T(n)={my(r=Vec(substvec(G(n),[x,y],[y,x]))); vector(#r-1, i, Vecrev(Pol(r[i+1]/y),i)) }
%o A369932 { my(A=T(12)); for(i=1, #A, print(A[i])) }
%Y A369932 Row sums are A369290.
%Y A369932 Column sums are A261919.
%Y A369932 Main diagonal is A008483.
%Y A369932 Cf. A342557 (connected), A123551 (without endpoints).
%K A369932 nonn,tabl
%O A369932 1,20
%A A369932 _Andrew Howroyd_, Feb 07 2024