A309936 Irregular triangle read by rows: T(n,k) is the number of unlabeled loopless multigraphs with n edges covering k vertices, n >= 1, 1 <= k <= 2*n.

0, 1, 0, 1, 1, 1, 0, 1, 2, 3, 1, 1, 0, 1, 3, 7, 6, 4, 1, 1, 0, 1, 4, 13, 17, 17, 8, 4, 1, 1, 0, 1, 6, 25, 44, 56, 41, 24, 9, 4, 1, 1, 0, 1, 7, 40, 101, 164, 158, 117, 57, 26, 9, 4, 1, 1, 0, 1, 9, 65, 216, 450, 562, 503, 315, 162, 64, 27, 9, 4, 1, 1

Offset

1,9

Comments

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

Links

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

Formula

T(n,k) = A192517(k,n) - A192517(k-1,n) for k > 1.

Example

Triangle begins:
  0, 1;
  0, 1, 1,  1;
  0, 1, 2,  3,   1,   1;
  0, 1, 3,  7,   6,   4,   1,   1;
  0, 1, 4, 13,  17,  17,   8,   4,  1,  1;
  0, 1, 6, 25,  44,  56,  41,  24,  9,  4, 1, 1;
  0, 1, 7, 40, 101, 164, 158, 117, 57, 26, 9, 4, 1, 1;
  ...

Prog

(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])) }

Crossrefs

Row sums are A050535.

Columns k=3..4 are A253186, A328652.

Cf. A191646, A192517, A327615.

Sequence in context: A004579 A081371 A321935 * A321900 A352195 A321895

Adjacent sequences: A309933 A309934 A309935 * A309937 A309938 A309939

Keyword

nonn,tabf

Author

Andrew Howroyd, Oct 23 2019

Status

approved