A240681 -id:A240681 - cp's OEIS frontend

A105599 Triangle read by rows: T(n, m) = number of forests with n nodes and m labeled trees. Also number of forests with exactly n - m edges on n labeled nodes.

1, 1, 1, 3, 3, 1, 16, 15, 6, 1, 125, 110, 45, 10, 1, 1296, 1080, 435, 105, 15, 1, 16807, 13377, 5250, 1295, 210, 21, 1, 262144, 200704, 76608, 18865, 3220, 378, 28, 1, 4782969, 3542940, 1316574, 320544, 55755, 7056, 630, 36, 1, 100000000, 72000000, 26100000, 6258000, 1092105, 143325, 14070, 990, 45, 1

Offset: 1

Washington Bomfim, Apr 14 2005; revised May 19 2005

Row sums equal A001858 (number of forests of labeled trees with n nodes).
Also the Bell transform of A000272(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016
The permutohedron (convex hull of permutations on 1,...,n in R^n) has Ehrhart polynomial Sum_{k=0..n-1} T(n,n-k) t^k. - Matthieu Josuat-Vergès, Mar 31 2018

			T(3, 2) = 3 because there are 3 such forests with 3 nodes and 2 trees.
Triangle begins:
      1;
      1,     1;
      3,     3,    1;
     16,    15,    6,    1;
    125,   110,   45,   10,   1;
   1296,  1080,  435,  105,  15,  1;
  16807, 13377, 5250, 1295, 210, 21, 1;

B. Bollobas, Graph Theory - An Introductory Course (Springer-Verlag, New York, 1979)

Alois P. Heinz, Rows n = 1..141, flattened
E. Britikov, Asymptotic number of forests from unrooted trees, Mat. Zametki (1988).
Mathoverflow, Is there a formula for the number of labeled forests with k components on n vertices?

Cf. A033185, A106240.
Rows reflected give A138464. - Alois P. Heinz, Sep 10 2008
Columns k=1-10 give: A000272, A083483, A239910, A240681, A240682, A240683, A240684, A240685, A240686, A240687.
T(2n,n) gives A302112.

Flat(List([1..11],n->List([1..n],m->(1/Factorial(m)*Sum([0..m],j->(-1/2)^j*Binomial(m,j)*Binomial(n-1,m+j-1)*n^(n-m-j)*Factorial(m+j)))))); # Muniru A Asiru, Apr 01 2018

T:= proc(n,m) option remember;
      if n<0 then 0
    elif n=m then 1
    elif m<1 or m>n then 0
    else add(binomial(n-1,j-1)*j^(j-2)*T(n-j,m-1), j=1..n-m+1)
      fi
    end:
seq(seq(T(n, m), m=1..n), n=1..12); # Alois P. Heinz, Sep 10 2008
# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> (n+1)^(n-1), 9); # Peter Luschny, Jan 27 2016

f[list_]:=Select[list,#>0&];Flatten[Map[f, Transpose[Table[t = Sum[n^(n - 2) x^n/n!, {n, 1, 20}];Drop[Range[0, 8]! CoefficientList[Series[t^k/k!, {x, 0, 8}], x],1], {k, 1, 8}]]]] (* Geoffrey Critzer, Nov 22 2011 *)
T[n_, m_] := Sum[(-1/2)^j*Binomial[m, j]*Binomial[n-1, m+j-1]*n^(n-m-j)*(m + j)!, {j, 0, m}]/m!; Table[T[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jan 09 2016, after Max Alekseyev *)
rows = 10;
t = Table[(n+1)^(n-1), {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

{ T(n,m) = sum(j=0,m, (-1/2)^j * binomial(m,j) * binomial(n-1,m+j-1) * n^(n-m-j)* (m+j)! )/m! } /* Max Alekseyev, Oct 08 2014 */

T(n,m) = Sum_{k=1..n-m+1} binomial(n-1,k-1)*k^(k-2)*T(n-k,m-1), T(n,0) = 0 if n > 0, T(0,0) = 1. - Vladeta Jovovic and Washington Bomfim
The value of T(n, m) can be calculated by the formula in Bollobas, pp. 172, exercise 44. Also T(n, m) = sum N/D over the partitions of n, 1*K(1) + 2*K(2) + ... + n*K(n), with exactly m parts, where N = n! * Product_{i = 1..n} i^( (i-2) * K(i) ) and D = Product_{i = 1..n} ( K(i)! * (i!)^K(i) ).
From Peter Bala, Aug 14 2012: (Start)
E.g.f.: A(x,t) := exp(t*F(x)) = 1 + t*x + (t + t^2)*x^2/2! + (3*t + 3*t^2 + t^3)*x^3/3! + ..., where F(x) = sum {n >= 1} n^(n-2)*x^n/n! is the e.g.f. for labeled trees (see A000272). The row polynomials R(n,t) are thus a sequence of binomial type polynomials.
Differentiating A(x,t) w.r.t. x yields A'(x,t) = t*A(x,t)*F'(x) leading to the recurrence equation for the row polynomials R(n,t) = t*sum {k = 0..n-1} (k+1)^(k-1)*binomial(n-1,k)*R(n-k-1,t) with R(0,t) = 1 and R(1,t) = t: the above recurrence for the table entries follows from this.
(End)
T(n,m) = (1/m!) * Sum_{j=0..m} (-1/2)^j * binomial(m,j) * binomial(n-1,m+j-1) * n^(n-m-j)* (m+j)!. Due to A. Renyi. - Max Alekseyev, Oct 08 2014
T(n,m) = (n!/m!)*Sum_{k_1+...+k_m=n, k_i>=1} Product_{j=1..m} k_j^(k_j-2)/k_j!. See Britikov reference. - Roland Vincze, Apr 18 2020

A138464 Triangle read by rows: T(n, k) is the number of forests on n labeled nodes with k edges. T(n, k) for n >= 1 and 0 <= k <= n-1.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 6, 15, 16, 1, 10, 45, 110, 125, 1, 15, 105, 435, 1080, 1296, 1, 21, 210, 1295, 5250, 13377, 16807, 1, 28, 378, 3220, 18865, 76608, 200704, 262144, 1, 36, 630, 7056, 55755, 320544, 1316574, 3542940, 4782969, 1, 45, 990, 14070, 143325, 1092105, 6258000, 26100000, 72000000, 100000000

Offset: 1

N. J. A. Sloane, May 09 2008

The rows of the triangle give the coefficients of the Ehrhart polynomials of integral Coxeter permutahedra of type A. These polynomials count lattice points in a dilated lattice polytope. For a definition see Ardila et al. (p. 1158), the generating functions of these polynomials for the classical root systems are given in theorem 5.2 (p. 1163). - Peter Luschny, May 01 2021

			Triangle begins:
[1]  1;
[2]  1,  1;
[3]  1,  3,   3;
[4]  1,  6,  15,   16;
[5]  1, 10,  45,  110,  125;
[6]  1, 15, 105,  435, 1080,  1296;
[7]  1, 21, 210, 1295, 5250, 13377, 16807;

Alois P. Heinz, Rows n = 1..141, flattened
Federico Ardila, Matthias Beck, and Jodi McWhirter, The arithmetic of Coxeter permutahedra, Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. 44(173):1152-1166, 2020.
John Riordan and N. J. A. Sloane, Correspondence, 1974

Row sums give A001858. Rightmost diagonal gives A000272. Cf. A136605.
Rows reflected give A105599. - Alois P. Heinz, Oct 28 2011
Cf. A088956.
Lower diagonals give: A083483, A239910, A240681, A240682, A240683, A240684, A240685, A240686, A240687. - Alois P. Heinz, Apr 11 2014
T(2n,n) gives A302112.
For Ehrhart polynomials of integral Coxeter permutahedra of classical type cf. this sequence (type A), A343805 (type B), A343806 (type C), A343807 (type D).

T:= proc(n) option remember; if n=0 then 0 else T(n-1) +n^(n-1) *x^n/n! fi end: TT:= proc(n) option remember; expand(T(n) -T(n)^2/2) end: f:= proc(k) option remember; if k=0 then 1 else unapply(f(k-1)(x) +x^k/k!, x) fi end: A:= proc(n,k) option remember; series(f(k)(TT(n)), x,n+1) end: aa:= (n,k)-> coeff(A(n,k), x,n) *n!: a:= (n,k)-> aa(n,n-k) -aa(n,n-k-1): seq(seq(a(n,k), k=0..n-1), n=1..10);  # Alois P. Heinz, Sep 02 2008
alias(W = LambertW): EhrA := exp(-W(-t*x)/t - W(-t*x)^2/(2*t)):
ser := series(EhrA, x, 12): cx := n -> n!*coeff(ser, x, n):
T := n -> seq(coeff(cx(n), t, k), k=0..n-1):
seq(T(n), n = 1..10); # Peter Luschny, Apr 30 2021

t[0, 0] = 1; t[n_ /; n >= 1, k_] /; (0 <= k <= n-1) := t[n, k] = Sum[(i+1)^(i-1)*Binomial[n-1, i]*t[n-i-1, k-i], {i, 0, k}]; t[, ] = 0; Table[t[n, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Jan 14 2014, after Peter Bala *)
gf := E^(-(ProductLog[-(t x)] (2 + ProductLog[-(t x)]))/(2 t));
ser := Series[gf, {x, 0, 12}]; cx[n_] := n! Coefficient[ser, x, n];
Table[CoefficientList[cx[n], t], {n, 1, 10}] // Flatten  (* Peter Luschny, May 01 2021 *)

From Peter Bala, Aug 14 2012: (Start)
T(n+1,k) = Sum_{i=0..k} (i+1)^(i-1)*binomial(n,i)*T(n-i,k-i) with T(0,0)=1.
Recurrence equation for row polynomials R(n,t): R(n,t) = Sum_{k=0..n-1} (k+1)^(k-1)*binomial(n-1,k)*t^k*R(n-k-1,t) with R(0,t) = R(1,t) = 1.
The production matrix for the row polynomials of the triangle is obtained from A088956 and starts:
    1    t
    1    1    t
    3    2    1    t
   16    9    3    1    t
  125   64   18    4    1   t
(End)
E.g.f.: exp( Sum_{n >= 1} n^(n-2)*t^(n-1)*x^n/n! ). - Peter Bala, Nov 08 2015
T(n, k) = [t^k] n! [x^n] exp(-W(-t*x)/t - W(-t*x)^2/(2*t)), where W denotes the Lambert function. - Peter Luschny, Apr 30 2021 [Typo corrected after note from Andrew Howroyd, Peter Luschny, Jun 20 2021]

More terms from Alois P. Heinz, Sep 02 2008

cp's OEIS Frontend

A105599 Triangle read by rows: T(n, m) = number of forests with n nodes and m labeled trees. Also number of forests with exactly n - m edges on n labeled nodes.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

Formula