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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.
4
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
1,4
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) = A290428(n,k) - A290428(n,k-1).
EXAMPLE
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;
...
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 A007717.
Columns k=2..3 are A024206, A327728.
Sequence in context: A123396 A225800 A176669 * A058280 A113185 A132069
KEYWORD
nonn,tabf
AUTHOR
Andrew Howroyd, Oct 23 2019
STATUS
approved