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.
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, 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, 162, 39, 6, 0, 0, 0, 0, 0, 5, 97, 515, 730, 332, 60, 9
Offset: 1
Examples
Triangle begins: 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, 9, 3; 0, 0, 0, 0, 1, 15, 32, 16, 4; 0, 0, 0, 0, 1, 12, 63, 76, 25, 5; ...
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
Crossrefs
Programs
-
PARI
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)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))} 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)} T(n)={my(r=Vec(substvec(G(n),[x,y],[y,x]))); vector(#r-1, i, Vecrev(Pol(r[i+1]/y),i)) } { my(A=T(12)); for(i=1, #A, print(A[i])) }