OFFSET
0,6
COMMENTS
This sequence excludes the graph consisting of a single isolated vertex without a loop. - Andrew Howroyd, Feb 02 2024
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1560 (rows 0..20)
FORMULA
E.g.f.: 1 - x + log(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 2 1
0 0 3 10 12 6 1
0 0 0 16 79 162 179 116 45 10 1
Row n = 3 counts the following loop-graphs (loops shown as singletons):
. . {12,13} {1,12,13} {1,2,12,13} {1,2,3,12,13} {1,2,3,12,13,23}
{12,23} {1,12,23} {1,2,12,23} {1,2,3,12,23}
{13,23} {1,13,23} {1,2,13,23} {1,2,3,13,23}
{2,12,13} {1,3,12,13} {1,2,12,13,23}
{2,12,23} {1,3,12,23} {1,3,12,13,23}
{2,13,23} {1,3,13,23} {2,3,12,13,23}
{3,12,13} {1,12,13,23}
{3,12,23} {2,3,12,13}
{3,13,23} {2,3,12,23}
{12,13,23} {2,3,13,23}
{2,12,13,23}
{3,12,13,23}
MATHEMATICA
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}], Length[Intersection@@s[[#]]]>0&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n], {1, 2}], {k}], Length[Union@@#]==n&&Length[csm[#]]<=1&]], {n, 0, 5}, {k, 0, Binomial[n+1, 2]}]
PROG
(PARI) T(n)={[Vecrev(p) | p<-Vec(serlaplace(1 - x + log(sum(j=0, n, (1 + y)^binomial(j+1, 2)*x^j/j!, O(x*x^n))))) ]}
{ my(A=T(6)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Feb 02 2024
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Jan 19 2024
STATUS
approved