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A369199
Irregular triangle read by rows where T(n,k) is the number of labeled loop-graphs covering n vertices with k edges.
33
1, 0, 1, 0, 1, 3, 1, 0, 0, 6, 17, 15, 6, 1, 0, 0, 3, 46, 150, 228, 206, 120, 45, 10, 1, 0, 0, 0, 45, 465, 1803, 3965, 5835, 6210, 4955, 2998, 1365, 455, 105, 15, 1, 0, 0, 0, 15, 645, 5991, 27364, 79470, 165555, 264050, 334713, 344526, 291200, 202860, 116190, 54258, 20349, 5985, 1330, 210, 21, 1
OFFSET
0,6
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1560 (rows 0..20)
FORMULA
E.g.f.: exp(-x) * (Sum_{j >= 0} (1 + y)^binomial(j+1, 2)*x^j/j!). - Andrew Howroyd, Feb 02 2024
EXAMPLE
Triangle begins:
1
0 1
0 1 3 1
0 0 6 17 15 6 1
0 0 3 46 150 228 206 120 45 10 1
Row n = 3 counts the following loop-graphs (loops shown as singletons):
{1,23} {1,2,3} {1,2,3,12} {1,2,3,12,13} {1,2,3,12,13,23}
{2,13} {1,2,13} {1,2,3,13} {1,2,3,12,23}
{3,12} {1,2,23} {1,2,3,23} {1,2,3,13,23}
{12,13} {1,3,12} {1,2,12,13} {1,2,12,13,23}
{12,23} {1,3,23} {1,2,12,23} {1,3,12,13,23}
{13,23} {1,12,13} {1,2,13,23} {2,3,12,13,23}
{1,12,23} {1,3,12,13}
{1,13,23} {1,3,12,23}
{2,3,12} {1,3,13,23}
{2,3,13} {1,12,13,23}
{2,12,13} {2,3,12,13}
{2,12,23} {2,3,12,23}
{2,13,23} {2,3,13,23}
{3,12,13} {2,12,13,23}
{3,12,23} {3,12,13,23}
{3,13,23}
{12,13,23}
MATHEMATICA
Table[Length[Select[Subsets[Subsets[Range[n], {1, 2}], {k}], Length[Union@@#]==n&]], {n, 0, 5}, {k, 0, Binomial[n+1, 2]}]
PROG
(PARI) T(n)={[Vecrev(p) | p<-Vec(serlaplace(exp(-x + O(x*x^n))*(sum(j=0, n, (1 + y)^binomial(j+1, 2)*x^j/j!)))) ]}
{ my(A=T(6)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Feb 02 2024
CROSSREFS
The version without loops is A054548.
This is the covering case of A084546.
Column sums are A173219.
Row sums are A322661, unlabeled A322700.
The connected case is A369195, without loops A062734.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts simple graphs; also loop-graphs if shifted left.
A006129 counts covering graphs, unlabeled A002494.
A368927 counts choosable loop-graphs, covering A369140.
A369141 counts non-choosable loop-graphs, covering A369142.
Sequence in context: A318973 A110517 A091925 * A034370 A144402 A264429
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Jan 18 2024
STATUS
approved