A005197 -id:A005197 - cp's OEIS frontend

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

Christian G. Bower

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

			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;

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

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

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

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 *)

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

A291336 Number F(n,h,t) of forests of t unlabeled rooted trees with n vertices such that h is the maximum of 0 and the tree heights; triangle of triangles F(n,h,t), n>=0, h=0..n, t=0..n-h, read by layers, then by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 2, 1, 0, 2, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 2, 1, 0, 4, 3, 1, 0, 3, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 3, 3, 2, 1, 0, 6, 8, 3, 1, 0, 8, 4, 1, 0, 4, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 3, 4, 3, 2, 1, 0, 10, 15, 9, 3, 1, 0, 18, 13, 4, 1, 0, 13, 5, 1, 0, 5, 1, 0, 1, 0

Offset: 0

Alois P. Heinz, Aug 22 2017

Elements in rows h=0 give A023531.
Positive elements in rows h=1 give A008284.
Positive row sums per layer (and - with a different offset - positive elements in column t=1) give A034781.
Positive column sums per layer give A033185.

			n h\t: 0 1 2 3 4 5 : A034781 : A033185   : A000081
-----+-------------+---------+-----------+--------
0 0  : 1           :         :           : 1
-----+-------------+---------+-----------+--------
1 0  : 0 1         :       1 : .         :
1 1  : 0           :         : 1         : 1
-----+-------------+---------+-----------+--------
2 0  : 0 0 1       :       1 : . .       :
2 1  : 0 1         :       1 : .         :
2 2  : 0           :         : 1 1       : 2
-----+-------------+---------+-----------+--------
3 0  : 0 0 0 1     :       1 : . . .     :
3 1  : 0 1 1       :       2 : . .       :
3 2  : 0 1         :       1 : .         :
3 3  : 0           :         : 2 1 1     : 4
-----+-------------+---------+-----------+--------
4 0  : 0 0 0 0 1   :       1 : . . . .   :
4 1  : 0 1 2 1     :       4 : . . .     :
4 2  : 0 2 1       :       3 : . .       :
4 3  : 0 1         :       1 : .         :
4 4  : 0           :         : 4 3 1 1   : 9
-----+-------------+---------+-----------+--------
5 0  : 0 0 0 0 0 1 :       1 : . . . . . :
5 1  : 0 1 2 2 1   :       6 : . . . .   :
5 2  : 0 4 3 1     :       8 : . . .     :
5 3  : 0 3 1       :       4 : . .       :
5 4  : 0 1         :       1 : .         :
5 5  : 0           :         : 9 6 3 1 1 : 20
-----+-------------+---------+-----------+--------

Alois P. Heinz, Layers n = 0..48, flattened

Cf. A000007, A000041, A000065, A000081, A001853, A005197, A008284, A023531, A033185, A034781, A291203, A291204, A291529.

b:= proc(n, i, t, h) option remember; expand(`if`(n=0 or h=0
       or i=1, x^(t*n), b(n, i-1, t, h)+add(x^(t*j)*binomial(
       b(i-1$2, 0, h-1)+j-1, j)*b(n-i*j, i-1, t, h), j=1..n/i)))
    end:
g:= (n, h)-> b(n$2, 1, h)-`if`(h=0, 0, b(n$2, 1, h-1)):
F:= (n, h, t)-> coeff(g(n, h), x, t):
seq(seq(seq(F(n, h, t), t=0..n-h), h=0..n), n=0..9);

b[n_, i_, t_, h_] := b[n, i, t, h] = Expand[If[n == 0 || h == 0
     || i == 1, x^(t*n), b[n, i-1, t, h] + Sum[x^(t*j)*Binomial[
     b[i-1, i-1, 0, h-1]+j-1, j]*b[n - i*j, i-1, t, h], {j, 1, n/i}]]];
g[n_, h_] := b[n, n, 1, h] - If[h == 0, 0, b[n, n, 1, h-1]];
F[n_, h_, t_] := Coefficient[g[n, h], x, t];
Table[Table[Table[F[n, h, t], {t, 0, n-h}], {h, 0, n}], {n, 0, 9}] //
Flatten (* Jean-François Alcover, Mar 10 2022, after Alois P. Heinz *)

Sum_{d=0..n} Sum_{i=0..d} F(n,i,d-i) = A000081(n+1).
Sum_{h=0..n} Sum_{t=0..n-h} t * F(n,h,t) = A005197(n).
Sum_{h=0..n} Sum_{t=0..n-h} (h+1) * F(n,h,t) = A001853(n+1) for n>0.
Sum_{t=0..n-1} F(n,1,t) = A000065(n) = A000041(n) - 1.
F(n,1,1) = 1 for n>1.
F(n,0,0) = A000007(n).

A291532 Number of trees in all forests of (unlabeled) rooted identity trees with n vertices.

Original entry on oeis.org

0, 1, 1, 3, 4, 9, 19, 40, 84, 186, 413, 922, 2082, 4733, 10831, 24928, 57648, 133923, 312393, 731328, 1717784, 4047111, 9561517, 22647521, 53770164, 127941813, 305046676, 728688803, 1743752229, 4179697971, 10034077377, 24123567285, 58076419495, 139996849639

Offset: 0

Alois P. Heinz, Aug 25 2017

nonn

			a(4) = 4:
:   o   :   o  o   :     o     :
:   |   :   |      :    / \    :
:   o   :   o      :   o   o   :
:   |   :   |      :   |       :
:   o   :   o      :   o       :
:   |   :          :           :
:   o   :          :           :
:       :          :           :

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A004111, A005197, A227774, A291529.

b:= proc(n, i, t) option remember; expand(`if`(n=0 or i=1,
       `if`(n<2, x^(t*n), 0), b(n, i-1, t)+add(binomial(
       b(i-1$2, 0), j)*x^(t*j)*b(n-i*j, i-1, t), j=1..n/i)))
    end:
a:= n-> (p-> add(i*coeff(p,x,i), i=1..degree(p)))(b(n$2, 1)):
seq(a(n), n=0..35);

b[n_, i_, t_] := b[n, i, t] = Expand[If[n == 0 || i == 1,
     If[n < 2, x^(t*n), 0], b[n, i - 1, t] + Sum[Binomial[
     b[i - 1, i - 1, 0], j]*x^(t*j)*b[n - i*j, i - 1, t], {j, 1, n/i}]]];
a[n_] := Function[p, Sum[i*Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][
     b[n, n, 1]];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Apr 29 2022, after Alois P. Heinz *)

a(n) = Sum_{k>=1} k * A227774(n+1,k).

a(n) = Sum_{h=0..n} Sum_{t=0..n-h} t * A291529(n,h,t).

A005199 a(n) = Sum_t t*F(n,t), where F(n,t) is the number of forests with n (unlabeled) nodes and exactly t trees, all of which are planted (that is, rooted trees in which the root has degree 1).

Original entry on oeis.org

0, 1, 1, 4, 6, 18, 35, 93, 214, 549, 1362, 3534, 9102, 23951, 63192, 168561, 451764, 1219290, 3305783, 9008027, 24643538, 67681372, 186504925, 515566016, 1429246490, 3972598378, 11068477743, 30908170493, 86488245455, 242481159915, 681048784377, 1916051725977, 5399062619966

Offset: 1

N. J. A. Sloane

nonn

The triangular array F(n,t) (analogous to A095133 for A005196 and A033185 for A005197) is A336087.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Washington Bomfim, Table of n, a(n) for n = 1..120
E. M. Palmer and A. J. Schwenk, On the number of trees in a random forest, J. Combin. Theory, B 27 (1979), 109-121.

Cf. A000081, A336087.

g(m) = {my(f); if(m==0, return(1)); f = vector(m+1); f[1]=1;
for(j=1, m, f[j+1]=1/j * sum(k=1, j, sumdiv(k,d, d * f[d]) * f[j-k+1])); f[m+1] };
global(max_n = 130); A000081 = vector(max_n, n, g(n-1));
F(n,t)={my(s=0, D, c, P_1); forpart(P_1 = n, D = Set(P_1); c = vector(#D);
for(k=1, #D, c[k] = #select(x->x == D[k], Vec(P_1)));
s += prod(k=1, #D, binomial( A000081[D[k]-1] + c[k] - 1, c[k]) )
,[2,n],[t,t]); s};
seq(n) = sum(t=1,n\2, t*F(n,t) ); \\   Washington Bomfim, Jul 08 2020

a(n) = Sum_{t=1, floor(n/2)}( t*F(n,t) ), where F(n,t) = Sum_{P_1(n,t)} (Product_{k=2..n} binomial(A000081(k-1) + c_k - 1, c_k)), where P_1(n, t) is the set of the partitions of n with t parts greater than one: 2*c_2 + ... + n*c_n = n; c_2, ..., c_n >= 0. - Washington Bomfim, Jul 08 2020

Definition clarified by N. J. A. Sloane, May 29 2012

cp's OEIS Frontend

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

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A291336 Number F(n,h,t) of forests of t unlabeled rooted trees with n vertices such that h is the maximum of 0 and the tree heights; triangle of triangles F(n,h,t), n>=0, h=0..n, t=0..n-h, read by layers, then by rows.

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A291532 Number of trees in all forests of (unlabeled) rooted identity trees with n vertices.

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A005199 a(n) = Sum_t t*F(n,t), where F(n,t) is the number of forests with n (unlabeled) nodes and exactly t trees, all of which are planted (that is, rooted trees in which the root has degree 1).

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