%I
%S 1,1,0,2,1,0,4,2,0,0,9,6,0,0,0,20,13,2,0,0,0,48,37,4,0,0,0,0,115,86,
%T 17,0,0,0,0,0,286,239,46,0,0,0,0,0,0,719,577,142,8,0,0,0,0,0,0,1842,
%U 1607,367,18,0,0,0,0,0,0,0,4766,4025,1136,76,0,0,0,0,0,0,0,0,12486,11185,2945,248,0,0,0,0,0,0,0,0,0
%N Triangle of the numbers of different forests with m rooted trees having distinct orders.
%C 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).
%H Alois P. Heinz, <a href="/A106236/b106236.txt">Rows n = 1..141, flattened</a>
%F a(n) = sum over the partitions of N: 1K1 + 2K2 + ... + NKN, with exactly m distinct parts, of Product_{i=1..N}binomial(A000081(i)+Ki1, 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}A000081(i). (Naturally, we do not consider the parts with multiplicity 0.)
%F G.f.: Product_{k>0} (1 + y*A000081(k)*x^k).  _Vladeta Jovovic_, May 14 2005
%e a(3) = 0 because m = 2 and (see comments) 3 < (2 + 3).
%e 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.
%p with(numtheory):
%p g:= proc(n) option remember; `if`(n<=1, n, (add(add(
%p d*g(d), d=divisors(j))*g(nj), j=1..n1))/(n1))
%p end:
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p expand(add(x^j*b(ni*j, i1)*binomial(g(i)+j1, j),
%p j=0..min(1, n/i)))))
%p end:
%p T:= n> (p> seq(coeff(p, x, i), i=1..n))(b(n$2)):
%p seq(T(n), n=1..14); # _Alois P. Heinz_, Jun 25 2014
%t g[n_] := g[n] = If[n <= 1, n, (Sum[Sum[d*g[d], {d, Divisors[j]}]*g[nj], {j, 1, n1}])/(n1)]; b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Expand[Sum[x^j*b[n  i*j, i1]*Binomial[g[i]+j1, 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 (* _JeanFrançois Alcover_, Feb 23 2015, after _Alois P. Heinz_ *)
%Y Cf. A000081, A106234.
%K nonn,tabl
%O 1,4
%A _Washington Bomfim_, Apr 28 2005
