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A106234 Triangle of the numbers of different forests with one or more isolated vertices. Those forests of rooted trees, have order N and m trees.

%I A106234 #16 Jan 14 2024 16:09:17
%S A106234 1,0,1,0,1,1,0,2,1,1,0,4,3,1,1,0,9,6,3,1,1,0,20,16,7,3,1,1,0,48,37,18,
%T A106234 7,3,1,1,0,115,96,44,19,7,3,1,1,0,286,239,117,46,19,7,3,1,1,0,719,622,
%U A106234 299,124,47,19,7,3,1,1,0,1842,1607,793,320,126,47,19,7,3,1,1
%N A106234 Triangle of the numbers of different forests with one or more isolated vertices. Those forests of rooted trees, have order N and m trees.
%C A106234 The unique tree with an isolated node has order one. For N > 1 and m > 1 there is at least one partition of N in m parts, with a part equal to 1, so a(n) > 0 when m > 1 and a(n) = 0 when m = 1 and N > 1.
%H A106234 Alois P. Heinz, <a href="/A106234/b106234.txt">Rows n = 1..141, flattened</a>
%F A106234 a(n) = sum over the partitions of N: 1K1 + 2K2 + ... + NKN, with exactly m parts and one or more parts equal to 1, of Product_{i=1..N} binomial(A000081(i)+Ki-1,Ki).
%e A106234 a(13) = 3 because 5 vertices can be partitioned in 3 trees in two ways: (1) one tree gets 3 nodes and the others get 1 each. (2) two trees get 2 nodes each and the other gets 1. Case (1) corresponds to 2 forests since A000081(3) = 2. Case (2) corresponds to one forest since A000081(2) = 1.
%e A106234 Triangle T(n,k) begins:
%e A106234   1;
%e A106234   0,   1;
%e A106234   0,   1,   1;
%e A106234   0,   2,   1,   1;
%e A106234   0,   4,   3,   1,  1;
%e A106234   0,   9,   6,   3,  1,  1;
%e A106234   0,  20,  16,   7,  3,  1, 1;
%e A106234   0,  48,  37,  18,  7,  3, 1, 1;
%e A106234   0, 115,  96,  44, 19,  7, 3, 1, 1;
%e A106234   0, 286, 239, 117, 46, 19, 7, 3, 1, 1;
%p A106234 with(numtheory):
%p A106234 g:= proc(n) option remember; `if`(n<=1, n, (add(add(
%p A106234       d*g(d), d=divisors(j))*g(n-j), j=1..n-1))/(n-1))
%p A106234     end:
%p A106234 b:= proc(n, i) option remember; `if`(n=0, 0, `if`(i=1,
%p A106234       x^n, expand(add(x^j*b(n-i*j, i-1)*
%p A106234       binomial(g(i)+j-1,j), j=0..n/i))))
%p A106234     end:
%p A106234 T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2)):
%p A106234 seq(T(n), n=1..14);  # _Alois P. Heinz_, Jun 25 2014
%t A106234 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)]; b[n_, i_] := b[n, i] = If[n == 0, 0, If[i == 1, x^n, Expand[ Sum[ x^j*b[n-i*j, i-1]*Binomial[g[i]+j-1, j], {j, 0,  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_, Mar 09 2015, after _Alois P. Heinz_ *)
%Y A106234 Cf. A033185, A105820.
%Y A106234 Row sums give A000081.
%K A106234 nonn,tabl
%O A106234 1,8
%A A106234 _Washington Bomfim_, Apr 26 2005