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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)).
5
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
0,4
LINKS
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.
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
EXAMPLE
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;
...
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 *)
PROG
(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
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.
Sequence in context: A187549 A261124 A100945 * A128751 A129168 A360733
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Nov 24 2007
STATUS
approved