A124503 Triangle read by rows: T(n,k) is the number of set partitions of the set {1,2,...,n} (or of any n-set) containing k blocks of size 3 (0<=k<=floor(n/3)).

1, 1, 2, 4, 1, 11, 4, 32, 20, 113, 80, 10, 422, 385, 70, 1788, 1792, 560, 8015, 9492, 3360, 280, 39435, 50640, 23100, 2800, 204910, 295020, 147840, 30800, 1144377, 1763300, 1044120, 246400, 15400, 6722107, 11278410, 7241520, 2202200, 200200, 41877722

Offset: 0

Author

Emeric Deutsch, Nov 14 2006

Keywords

• nonn
• tabf

Comments

Row n contains 1+floor(n/3) terms. Row sums yield the Bell numbers (A000110). T(n,0)=A124504(n). Sum(k*T(n,k), k=0..floor(n/3))=A105480(n+1).

Examples

			T(4,1)=4 because we have 1|234, 134|2, 124|3 and 123|4.
Triangle starts:
    1;
    1;
    2;
    4,   1;
   11,   4;
   32,  20;
  113,  80, 10;
  422, 385, 70;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..250, flattened

Crossrefs

Cf. A000110, A124504, A105480, A355144.

T(3n,n) gives A025035.

Programs

Maple

G:=exp(exp(z)-1+(t-1)*z^3/6): Gser:=simplify(series(G,z=0,17)): for n from 0 to 14 do P[n]:=sort(n!*coeff(Gser,z,n)) od: for n from 0 to 14 do seq(coeff(P[n],t,k),k=0..floor(n/3)) od; # yields sequence in triangular form
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
      b(n-i*j, i-1)*`if`(i=3, x^j, 1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..15);  # Alois P. Heinz, Mar 08 2015

Mathematica

nn = 8; k = 3; Range[0, nn]! CoefficientList[Series[Exp[Exp[x] - 1 + (y - 1) x^k/k!], {x, 0, nn}], {x, y}] // Grid (* Geoffrey Critzer, Aug 26 2012 *)

Formula

E.g.f.: G(t,z) = exp(exp(z)-1+(t-1)z^3/6).