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 A106235 Triangle of the numbers of different forests of m rooted trees of smallest order 2, i.e., without isolated vertices. 1
 0, 1, 0, 2, 0, 0, 4, 1, 0, 0, 9, 2, 0, 0, 0, 20, 7, 1, 0, 0, 0, 48, 17, 2, 0, 0, 0, 0, 115, 48, 7, 1, 0, 0, 0, 0, 286, 124, 21, 2, 0, 0, 0, 0, 0, 719, 336, 60, 7, 1, 0, 0, 0, 0, 0, 1842, 888, 171, 21, 2, 0, 0, 0, 0, 0, 0, 4766, 2393, 488, 65, 7, 1, 0, 0, 0, 0, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Forests of order N with m components, m > floor(N/2) must contain an isolated vertex since it is impossible to partition N vertices in floor(N/2) + 1 or more trees without giving only one vertex to a tree. A033185(n) = A106235(n) + A106234(n). LINKS Alois P. Heinz, Rows n = 1..141, flattened FORMULA a(n) = sum over the partitions of N: 1K1 + 2K2 + ... + NKN, with exactly m parts and no part equal to 1, of Product_{i=1..N} binomial(A000081(i)+Ki-1, Ki). EXAMPLE a(12)=2 because 5 nodes can be partitioned in two trees only in a way: one tree gets 3 nodes and the other tree gets 2. Since A000081(3) = 2 and A000081(2)=1, there are two forests. Triangle T(n,k) begins:     0;     1,   0;     2,   0,  0;     4,   1,  0, 0;     9,   2,  0, 0, 0;    20,   7,  1, 0, 0, 0;    48,  17,  2, 0, 0, 0, 0;   115,  48,  7, 1, 0, 0, 0, 0;   286, 124, 21, 2, 0, 0, 0, 0, 0;   719, 336, 60, 7, 1, 0, 0, 0, 0, 0; 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: 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(g(i)+j-1, j), j=0..n/i))))     end: T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n\$2)): seq(T(n), n=1..14);  # Alois P. Heinz, Jun 25 2014 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)]; 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[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, Nov 05 2015, after Alois P. Heinz *) CROSSREFS Cf. A033185, A106234. Sequence in context: A136334 A155039 A238858 * A288098 A118965 A252729 Adjacent sequences:  A106232 A106233 A106234 * A106236 A106237 A106238 KEYWORD nonn,tabl AUTHOR Washington Bomfim, Apr 26 2005 STATUS approved

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Last modified August 14 07:04 EDT 2020. Contains 336477 sequences. (Running on oeis4.)