|
|
A368983
|
|
Number of connected graphs with loops (symmetric relations) on n unlabeled vertices with n edges.
|
|
8
|
|
|
1, 1, 1, 3, 6, 14, 33, 81, 204, 526, 1376, 3648, 9792, 26485, 72233, 198192, 546846, 1515687, 4218564, 11782427, 33013541, 92759384, 261290682, 737688946, 2086993034, 5915398230, 16795618221, 47763406249, 136028420723, 387928330677, 1107692471888, 3166613486137
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,4
|
|
COMMENTS
|
The graphs considered here can have loops but not parallel edges.
|
|
LINKS
|
|
|
FORMULA
|
|
|
EXAMPLE
|
Representatives of the a(3) = 3 graphs are:
{{1,2}, {1,3}, {2,3}},
{{1}, {1,2}, {1,3}},
{{1}, {1,2}, {2,3}}.
|
|
PROG
|
(PARI) \\ TreeGf gives gf of A000081.
TreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
seq(n)={my(t=TreeGf(n)); my(g(e)=subst(t + O(x*x^(n\e)), x, x^e) + O(x*x^n)); Vec(1 + (sum(d=1, n, eulerphi(d)/d*log(1/(1-g(d)))) + ((1+g(1))^2/(1-g(2))-1)/2 - (g(1)^2 + g(2)))/2)}
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|