A327615 Irregular triangle read by rows: T(n,k) is the number of unlabeled multigraphs with loops allowed and n edges covering k vertices, n >= 1, 1 <= k <= 2*n.

1, 1, 1, 3, 2, 1, 1, 5, 8, 6, 2, 1, 1, 8, 19, 25, 16, 7, 2, 1, 1, 11, 40, 73, 73, 47, 19, 7, 2, 1, 1, 15, 77, 194, 263, 232, 133, 58, 20, 7, 2, 1, 1, 19, 132, 454, 835, 951, 719, 397, 164, 61, 20, 7, 2, 1, 1, 24, 217, 984, 2385, 3507, 3365, 2306, 1177, 490, 175, 62, 20, 7, 2, 1

Offset: 1

Author

Andrew Howroyd, Oct 23 2019

Keywords

• nonn
• tabf

Comments

Covering k vertices means there are no vertices of degree zero.

Examples

			Triangle begins:
  1,  1;
  1,  3,   2,   1;
  1,  5,   8,   6,   2,   1;
  1,  8,  19,  25,  16,   7,   2,   1;
  1, 11,  40,  73,  73,  47,  19,   7,   2,  1;
  1, 15,  77, 194, 263, 232, 133,  58,  20,  7,  2, 1;
  1, 19, 132, 454, 835, 951, 719, 397, 164, 61, 20, 7, 2, 1;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..650 (rows 1..25)

Crossrefs

Row sums are A007717.

Columns k=2..3 are A024206, A327728.

Cf. A136564, A290428, A309936, A322115.

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)))}
C(n,m)={my(s=O(x*x^m)); forpart(p=n, s+=permcount(p)/edges(p, i->1-x^i+O(x*x^m))); Col(s/n!)}
T(m) = {my(n=2*m, A=Mat(vector(n+1, n, C(n-1,m)))); A[2..m+1, 2..n+1]-A[2..m+1, 1..n]}
{ my(A=T(8)); for(n=1, matsize(A)[1], print(A[n, 1..2*n])) }

Formula

T(n,k) = A290428(n,k) - A290428(n,k-1).