A370063 Triangle read by rows: T(n,k) is the number of unlabeled loopless multigraphs without endpoints with n edges covering k vertices, 0 <= k <= n.

1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 2, 0, 0, 1, 3, 5, 2, 0, 0, 1, 5, 14, 10, 4, 0, 0, 1, 6, 25, 33, 18, 4, 0, 0, 1, 8, 46, 96, 90, 31, 7, 0, 0, 1, 10, 75, 227, 330, 194, 52, 8, 0, 0, 1, 12, 117, 494, 1033, 962, 416, 82, 12, 0, 0, 1, 14, 173, 982, 2847, 3908, 2591, 800, 128, 14

Offset: 0

Author

Andrew Howroyd, Feb 08 2024

Keywords

• nonn
• tabl

Comments

An endpoint is a vertex that appears in only one edge. Equivalently, the degree of every vertex >= 2.

Examples

			Triangle begins:
  1;
  0, 0;
  0, 0, 1;
  0, 0, 1,  1;
  0, 0, 1,  2,   2;
  0, 0, 1,  3,   5,   2;
  0, 0, 1,  5,  14,  10,    4;
  0, 0, 1,  6,  25,  33,   18,   4;
  0, 0, 1,  8,  46,  96,   90,  31,   7;
  0, 0, 1, 10,  75, 227,  330, 194,  52,  8;
  0, 0, 1, 12, 117, 494, 1033, 962, 416, 82, 12;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Crossrefs

Row sums are A307316.

Main diagonal is A002865.

Cf. A369932.

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, i, Vecrev(Pol(r[i]), i)) }
{ my(A=T(10)); for(i=1, #A, print(A[i])) }