A133399 Triangle T(n,k)=number of forests of labeled rooted trees with n nodes, containing exactly k trees of height one, all others having height zero (n>=0, 0<=k<=floor(n/2)).

1, 1, 1, 2, 1, 9, 1, 28, 12, 1, 75, 120, 1, 186, 750, 120, 1, 441, 3780, 2100, 1, 1016, 16856, 21840, 1680, 1, 2295, 69552, 176400, 45360, 1, 5110, 272250, 1224720, 705600, 30240, 1, 11253, 1026300, 7692300, 8316000, 1164240, 1, 24564, 3762132, 45018600

Offset: 0

Author

Alois P. Heinz, Nov 24 2007

Keywords

• nonn
• tabf

Examples

			Triangle begins:
  1;
  1;
  1,     2;
  1,     9;
  1,    28,     12;
  1,    75,    120;
  1,   186,    750,     120;
  1,   441,   3780,    2100;
  1,  1016,  16856,   21840,   1680;
  1,  2295,  69552,  176400,  45360;
  1,  5110, 272250, 1224720, 705600, 30240;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened
• A. P. Heinz, Finding Two-Tree-Factor Elements of Tableau-Defined Monoids in Time O(n^3), Ed. S. G. Akl, F. Fiala, W. W. Koczkodaj: Advances in Computing and Information, ICCI90 Niagara Falls, LNCS 468, Springer-Verlag (1990), pp. 120-128.

Crossrefs

Columns k=1,2 give: A058877, A133386.

Row sums give: A000248.

T(2n,n) = A001813(n), T(2n+1,n) = A002691(n).

Reading the table by diagonals gives triangle A198204. - Peter Bala, Jul 31 2012

Cf. A235596.

Programs

Magma

/* As triangle */ [[Binomial(n,k)*Factorial(k)*StirlingSecond(n-k+1,k+1): k in [0..Floor(n/2)]]: n in [0.. 15]]; // Vincenzo Librandi, Jun 06 2019

Maple

T:= (n,k)-> binomial(n,k)*k!*Stirling2(n-k+1,k+1): for n from 0 to 10 do lprint(seq(T(n, k), k=0..floor(n/2))) od;

Mathematica

nn=12;f[list_]:=Select[list,#>0&];Map[f,Range[0,nn]!CoefficientList[ Series[Exp[y x (Exp[x]-1)] Exp[x],{x,0,nn}],{x,y}]]//Grid (* Geoffrey Critzer, Feb 09 2013 *)
t[n_, k_] := Binomial[n, k]*k!*StirlingS2[n-k+1, k+1]; Table[t[n, k], {n, 0, 12}, {k, 0, n/2}] // Flatten (* Jean-François Alcover, Dec 19 2013 *)

Formula

T(n,k) = C(n,k) * k! * stirling2(n-k+1,k+1).

E.g.f.: exp(y*x*(exp(x)-1))*exp(x). - Geoffrey Critzer, Feb 09 2013

Sum_{k=1..floor(n/2)} T(n,k) = A235596(n+1). - Alois P. Heinz, Jun 21 2019