A033185 Rooted tree triangle read by rows: a(n,k) = number of forests with n nodes and k rooted trees.

1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 9, 6, 3, 1, 1, 20, 16, 7, 3, 1, 1, 48, 37, 18, 7, 3, 1, 1, 115, 96, 44, 19, 7, 3, 1, 1, 286, 239, 117, 46, 19, 7, 3, 1, 1, 719, 622, 299, 124, 47, 19, 7, 3, 1, 1, 1842, 1607, 793, 320, 126, 47, 19, 7, 3, 1, 1, 4766, 4235, 2095, 858, 327, 127, 47, 19, 7, 3, 1, 1

Offset: 1

Author

Christian G. Bower

Keywords

• nonn
• tabl

Comments

Leading column: A000081, rows sums: A000081 shifted.

Also, number of multigraphs of k components, n nodes, and no cycles except one loop in each component. See link below to have a picture showing the bijection between rooted forests and multigraphs of this kind. - Washington Bomfim, Sep 04 2010

Number of rooted trees with n+1 nodes and degree of the root is k.- Michael Somos, Aug 20 2018

Examples

			Triangle begins:
     1;
     1,    1;
     2,    1,   1;
     4,    3,   1,   1;
     9,    6,   3,   1,   1;
    20,   16,   7,   3,   1,  1;
    48,   37,  18,   7,   3,  1,  1;
   115,   96,  44,  19,   7,  3,  1,  1;
   286,  239, 117,  46,  19,  7,  3,  1,  1;
   719,  622, 299, 124,  47, 19,  7,  3,  1,  1;
  1842, 1607, 793, 320, 126, 47, 19,  7,  3,  1,  1;
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened
• W. Bomfim, Bijection between rooted forests and multigraphs without cycles except one loop in each connected component.
• R. J. Mathar, Topologically distinct sets of non-intersecting circles in the plane, arXiv:1603.00077 [math.CO] (2016), Table 2.
• Index entries for sequences related to rooted trees
• Index entries for sequences related to trees

Crossrefs

Cf. A000081, A005197, A106240, A181360, A027852 (2nd column), A000226 (3rd column), A029855 (4th column), A336087.

Programs

Maple

with(numtheory):
t:= proc(n) option remember; local d, j; `if` (n<=1, n,
      (add(add(d*t(d), d=divisors(j))*t(n-j), j=1..n-1))/(n-1))
    end:
b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
      `if`(min(i, p)<1, 0, add(b(n-i*j, i-1, p-j) *
       binomial(t(i)+j-1, j), j=0..min(n/i, p)))))
    end:
a:= (n, k)-> b(n, n, k):
seq(seq(a(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Aug 20 2012

Mathematica

nn=10;f[x_]:=Sum[a[n]x^n,{n,0,nn}];sol=SolveAlways[0 == Series[f[x]-x Product[1/(1-x^i)^a[i],{i,1,nn}],{x,0,nn}],x];a[0]=0;g=Table[a[n],{n,1,nn}]/.sol//Flatten;h[list_]:=Select[list,#>0&];Map[h,Drop[CoefficientList[Series[x Product[1/(1-y x^i)^g[[i]],{i,1,nn}],{x,0,nn}],{x,y}],2]]//Grid  (* Geoffrey Critzer, Nov 17 2012 *)
t[1] = 1; t[n_] := t[n] = Module[{d, j}, Sum[Sum[d*t[d], {d, Divisors[j]}]*t[n-j], {j, 1, n-1}]/(n-1)]; b[1, 1, 1] = 1; b[n_, i_, p_] := b[n, i, p] = If[p>n, 0, If[n == 0, 1, If[Min[i, p]<1, 0, Sum[b[n-i*j, i-1, p-j]*Binomial[t[i]+j-1, j], {j, 0, Min[n/i, p]}]]]]; a[n_, k_] := b[n, n, k]; Table[a[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 13 2014, after Alois P. Heinz *)

Formula

G.f.: 1/Product_{i>=1} (1-x*y^i)^A000081(i). - Vladeta Jovovic, Apr 28 2005

a(n, k) = sum over the partitions of n, 1M1 + 2M2 + ... + nMn, with exactly k parts, of Product_{i=1..n} binomial(A000081(i)+Mi-1, Mi). - Washington Bomfim, May 12 2005