|
|
A317057
|
|
a(n) is the number of time-dependent assembly trees satisfying the connected gluing rule for a cycle on n vertices.
|
|
5
|
|
|
1, 1, 4, 23, 166, 1437, 14512, 167491, 2174746, 31374953, 497909380, 8619976719, 161667969646, 3265326093109, 70663046421208, 1631123626335707, 40004637435452866, 1038860856732399105, 28476428717448349996
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
COMMENTS
|
A time-dependent assembly tree for a connected graph G=(V, E) on n vertices is a rooted tree, each node of which is label a subset U of V and a nonnegative integer i such that:
1) each internal node has at least two children,
2) there are leaves labeled (v, 0) for each vertex v in V,
3) the label on the root is (V, m) for 1 <= m <= n-1,
4) for each node (U, i) with i < m, U is the union of the {u} for the children (u, 0) of (U, i),
5) if (U, i) and (U', i') are adjacent nodes with U a subset of U', then i < i',
6) for each 0 <= i <= m, there exists a node (U, i) with U a subset of V.
A time-dependent assembly tree is said to satisfy the connected gluing rule if each internal vertex v of G, the graph induced by the vertices in the labels is connected.
|
|
LINKS
|
|
|
FORMULA
|
a(n) = 1 + Sum_{j=2..n-1} binomial(n, j)*a(j), a(1)=a(2)=1.
E.g.f.: (x - x*e^x + e^x - 1)/(2 - e^x).
|
|
MAPLE
|
option remember;
if n <=2 then
1;
else
1+add(binomial(n, j)*procname(j), j=2..n-1) ;
end if;
end proc:
|
|
MATHEMATICA
|
Nest[Function[{a, n}, Append[a, 1 + Sum[Binomial[n, j] a[[j]], {j, 2, n - 1}]]][#, Length@ # + 1] &, {1, 1}, 17] (* Michael De Vlieger, Jul 26 2018 *)
|
|
PROG
|
(Sage)
@cached_function
def TimeDepenConCycle(n):
if (n==1):
return 1
elif (n==2):
return 1
else:
return sum([binomial(n, j)*TimeDepenConCycle(j) for j in range(2, n)])+1
print(', '.join(str(TimeDepenConCycle(i)) for i in range(1, 20)))
(GAP) a:=[1, 1];; for n in [3..20] do a[n]:=1+Sum([2..n-1], j->Binomial(n, j)*a[j]); od; a; # Muniru A Asiru, Jul 25 2018
(PARI) lista(nn) = my(v = vector(nn)); for (n=1, nn, if (n<=2, v[n] = 1, v[n] = 1 + sum(j=2, n-1, binomial(n, j)*v[j]))); v; \\ Michel Marcus, Aug 08 2018
|
|
CROSSREFS
|
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|