A124498 Triangle read by rows: T(n,k) is the number of set partitions of the set {1,2,...,n} containing k blocks of size 2 (0 <= k <= floor(n/2)).

1, 1, 1, 1, 2, 3, 6, 6, 3, 17, 20, 15, 53, 90, 45, 15, 205, 357, 210, 105, 871, 1484, 1260, 420, 105, 3876, 7380, 6426, 2520, 945, 18820, 39195, 33390, 18900, 4725, 945, 99585, 213180, 202950, 117810, 34650, 10395, 558847, 1242120, 1293435, 734580, 311850

Offset: 0

Author

Emeric Deutsch, Nov 05 2006

Keywords

• nonn
• tabf

Comments

Row n contains 1+floor(n/2) terms. Row sums yield the Bell numbers A000110. T(n,0)=A097514(n). Sum(k*T(n,k), k=0..floor(n/2))=A105479(n+1).

Examples

			T(4,1)=6 because we have 12|3|4, 13|2|4, 14|2|3, 1|23|4, 1|24|3 and 1|2|34.
Triangle T(n,k) begins:
:      1;
:      1;
:      1,       1;
:      2,       3;
:      6,       6,       3;
:     17,      20,      15;
:     53,      90,      45,     15;
:    205,     357,     210,    105;
:    871,    1484,    1260,    420,    105;
:   3876,    7380,    6426,   2520,    945;
:  18820,   39195,   33390,  18900,   4725,   945;
:  99585,  213180,  202950, 117810,  34650, 10395;
: 558847, 1242120, 1293435, 734580, 311850, 62370, 10395;
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened

Crossrefs

Cf. A000110, A097514, A105479, A124503.

T(2n,n) gives A001147.

Programs

Maple

G:=exp(exp(z)-1+(t-1)*z^2/2): Gser:=simplify(series(G,z=0,16)): for n from 0 to 13 do P[n]:=sort(n!*coeff(Gser,z,n)) od: for n from 0 to 13 do seq(coeff(P[n],t,k),k=0..floor(n/2)) 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=2, 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

d=Exp[Exp[x]-x^2/2!-1]; f[list_] := Select[list,#>0&]; Map[f, Transpose[Table[Range[0,12]! CoefficientList[Series[ x^(2k)/(k! 2!^k) *d, {x,0,12}], x], {k,0,5}]]]//Flatten (* Geoffrey Critzer, Nov 30 2011 *)

Formula

E.g.f.: exp(exp(z)-1+(t-1)z^2/2).

Generally the e.g.f. for set partitions containing k blocks of size p is: G(z,t) = exp(exp(z)-1+(t-1)z^p/p!) - Geoffrey Critzer, Nov 30 2011