|
| |
|
|
A106237
|
|
Triangle of the numbers of different forests with m trees having distinct orders.
|
|
2
|
|
|
|
1, 1, 0, 1, 1, 0, 2, 1, 0, 0, 3, 3, 0, 0, 0, 6, 5, 1, 0, 0, 0, 11, 11, 2, 0, 0, 0, 0, 23, 20, 5, 0, 0, 0, 0, 0, 47, 46, 11, 0, 0, 0, 0, 0, 0, 106, 93, 26, 2, 0, 0, 0, 0, 0, 0, 235, 216, 58, 3, 0, 0, 0, 0, 0, 0, 0, 551, 467, 139, 12, 0, 0, 0, 0, 0, 0, 0, 0, 1301, 1121, 307, 29, 0, 0, 0, 0, 0, 0, 0, 0, 0
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,7
|
|
|
COMMENTS
|
a(n) = 0 if and only if n < m + (((1+m)*m - 1)^2 -1)/8, where m is the number of trees in the forests counted by a(n).
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = sum over the partitions of N: 1K1 + 2K2 + ... + NKN, with exactly m distinct parts, of Product_{i=1..N}binomial(A000055(i)+Ki-1, Ki). Because all the multiplicities of the parts of the considered partitions are 1, or 0, we can simplify the formula to a(n) = sum over the partitions of N with exactly m distinct parts, of Product_{i=1..N}A000055(i). (Naturally, we do not consider the parts with multiplicity 0.)
|
|
|
EXAMPLE
|
a(3) = 0 because m = 2 and (see comments) 3 < (2 + 3).
a(4) > 0 because m = 1. Note that (((1+m)*m - 1)^2 - 1)/8 = 0, if m = 1. It is clear that n >= m.
|
|
|
MAPLE
|
with(numtheory):
g:= proc(n) option remember; `if`(n<=1, n, (add(add(
d*g(d), d=divisors(j))*g(n-j), j=1..n-1))/(n-1))
end:
h:= n-> `if`(n=0, 1, g(n) -(add(g(k) *g(n-k), k=0..n)
-`if`(irem(n, 2)=0, g(n/2), 0))/2):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
expand(add(x^j*b(n-i*j, i-1)*binomial(h(i)+j-1, j),
j=0..min(1, n/i)))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2)):
|
|
|
MATHEMATICA
|
g[n_] := g[n] = If[n <= 1, n, Sum[Sum[d*g[d], {d, Divisors[j]}]*g[n-j], {j, 1, n-1}]/(n-1)]; h[n_] := If[n == 0, 1, g[n] - (Sum[g[k]*g[n-k], {k, 0, n}] - If[Mod[n, 2] == 0, g[n/2], 0])/2]; b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Expand[ Sum[x^j*b[n-i*j, i-1]*Binomial[h[i]+j-1, j], {j, 0, Min[1, n/i]}]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, n]]; Table[T[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jan 28 2015, after Alois P. Heinz *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
