|
|
A301343
|
|
Regular triangle where T(n,k) is the number of planted achiral (or generalized Bethe) trees with n nodes and k leaves.
|
|
7
|
|
|
1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 1, 1, 1, 0, 1, 3, 2, 2, 1, 1, 0, 1, 3, 2, 2, 1, 1, 1, 0, 1, 4, 2, 4, 1, 2, 1, 1, 0, 1, 4, 3, 4, 1, 3, 1, 1, 1, 0, 1, 5, 3, 6, 2, 4, 1, 2, 1, 1, 0, 1, 5, 3, 6, 2, 4, 1, 2, 1, 1, 1, 0, 1, 6, 4, 9, 2, 7, 1, 4, 2, 2, 1, 1, 0
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,12
|
|
LINKS
|
|
|
FORMULA
|
T(n,1) = 1, T(n,k) = 0 if n <= k, otherwise T(n,k) = Sum_{d|k} T(n - k, d).
|
|
EXAMPLE
|
Triangle begins:
1
1 0
1 1 0
1 1 1 0
1 2 1 1 0
1 2 1 1 1 0
1 3 2 2 1 1 0
1 3 2 2 1 1 1 0
1 4 2 4 1 2 1 1 0
1 4 3 4 1 3 1 1 1 0
1 5 3 6 2 4 1 2 1 1 0
The T(9,4) = 4 planted achiral trees: (((((oooo))))), ((((oo)(oo)))), (((oo))((oo))), ((o)(o)(o)(o)).
|
|
MATHEMATICA
|
tri[n_, k_]:=If[k===1, 1, If[k>=n, 0, Sum[tri[n-k, d], {d, Divisors[k]}]]];
Table[tri[n, k], {n, 10}, {k, n}]
|
|
CROSSREFS
|
Row sums are A003238. A version without the zeroes or first row is A214575.
Cf. A000081, A003238, A004111, A032305, A055277, A214577, A273873, A289078, A289079, A290689, A291443, A298422, A298426, A301342, A301344, A301345.
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|