A245797
The number of labeled graphs of n vertices that have endpoints, where an endpoint is a vertex with degree 1.
Original entry on oeis.org
0, 1, 6, 49, 710, 19011, 954184, 90154415, 16108626420, 5481798833245, 3582369649269620, 4532127781040045649, 11177949079089720090800, 54050029251399545975868271, 514598463471970554205910304780, 9677402372862708729859372687791391
Offset: 1
The generalization to set-systems is
A327228.
BII-numbers of set-systems with minimum degree 1 are
A327105.
Cf.
A001187,
A006129,
A059166,
A059167,
A100743,
A136284,
A327079,
A327098,
A327103,
A327229,
A327230.
-
m = 16;
egf = Exp[x^2/2]*Sum[2^Binomial[n, 2]*(x/Exp[x])^n/n!, {n, 0, m}];
A059167[n_] := SeriesCoefficient[egf, {x, 0, n}]*n!;
a[n_] := 2^(n(n-1)/2) - A059167[n];
Array[a, m] (* Jean-François Alcover, Feb 23 2019 *)
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Min@@Length/@Split[Sort[Join@@#]]==1&]],{n,0,5}] (* Gus Wiseman, Sep 11 2019 *)
A369928
Triangle read by rows: T(n,k) is the number of simple graphs on n labeled vertices with k edges and without endpoints, n >= 0, 0 <= k <= n*(n-1)/2.
Original entry on oeis.org
1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 4, 3, 6, 1, 1, 0, 0, 10, 15, 42, 90, 100, 45, 10, 1, 1, 0, 0, 20, 45, 162, 595, 1590, 3075, 3655, 2703, 1335, 455, 105, 15, 1, 1, 0, 0, 35, 105, 462, 2310, 9495, 32130, 85365, 166341, 231861, 237125, 184380, 111870, 53634, 20307, 5985, 1330, 210, 21, 1
Offset: 0
Triangle begins:
[0] 1;
[1] 1;
[2] 1, 0;
[3] 1, 0, 0, 1;
[4] 1, 0, 0, 4, 3, 6, 1;
[5] 1, 0, 0, 10, 15, 42, 90, 100, 45, 10, 1;
[6] 1, 0, 0, 20, 45, 162, 595, 1590, 3075, 3655, 2703, 1335, 455, 105, 15, 1;
-
\\ row(n) gives n-th row as vector.
row(n)={my(A=x/exp(x*y + O(x*x^n))); Vecrev(polcoef(serlaplace(exp(y*x^2/2 + O(x*x^n)) * sum(k=0, n, (1 + y)^binomial(k, 2)*A^k/k!)), n), 1 + binomial(n,2))}
{ for(n=0, 6, print(row(n))) }
A240168
T(n,k) is the number of unlabeled graphs of n vertices and k edges that have endpoints, where an endpoint is a vertex with degree 1.
Original entry on oeis.org
0, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 5, 4, 2, 1, 1, 2, 4, 8, 13, 15, 16, 11, 5, 2, 1, 1, 2, 4, 9, 19, 35, 55, 75, 83, 72, 51, 29, 13, 5, 2, 1, 1, 2, 4, 10, 22, 50, 105, 196, 338, 511, 649, 695, 627, 473, 304, 172, 83, 35, 14, 5, 2, 1
Offset: 1
First few rows of irregular triangle are:
..0
..1
..1....1
..1....2....2....1
..1....2....3....5....4....2....1
..1....2....4....8...13...15...16...11....5....2....1
..1....2....4....9...19...35...55...75...83...72...51...29...13....5....2....1
...
Showing 1-3 of 3 results.
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