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%I #21 Apr 08 2016 07:03:15
%S 1,1,1,1,2,3,6,6,3,17,20,15,53,90,45,15,205,357,210,105,871,1484,1260,
%T 420,105,3876,7380,6426,2520,945,18820,39195,33390,18900,4725,945,
%U 99585,213180,202950,117810,34650,10395,558847,1242120,1293435,734580,311850
%N 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)).
%C 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).
%H Alois P. Heinz, <a href="/A124498/b124498.txt">Rows n = 0..200, flattened</a>
%F E.g.f.: exp(exp(z)-1+(t-1)z^2/2).
%F 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
%e 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.
%e Triangle T(n,k) begins:
%e : 1;
%e : 1;
%e : 1, 1;
%e : 2, 3;
%e : 6, 6, 3;
%e : 17, 20, 15;
%e : 53, 90, 45, 15;
%e : 205, 357, 210, 105;
%e : 871, 1484, 1260, 420, 105;
%e : 3876, 7380, 6426, 2520, 945;
%e : 18820, 39195, 33390, 18900, 4725, 945;
%e : 99585, 213180, 202950, 117810, 34650, 10395;
%e : 558847, 1242120, 1293435, 734580, 311850, 62370, 10395;
%p 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
%p # second Maple program:
%p with(combinat):
%p b:= proc(n, i) option remember; expand(`if`(n=0, 1,
%p `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
%p b(n-i*j, i-1)*`if`(i=2, x^j, 1), j=0..n/i))))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
%p seq(T(n), n=0..15); # _Alois P. Heinz_, Mar 08 2015
%t 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 *)
%Y Cf. A000110, A097514, A105479, A124503.
%Y T(2n,n) gives A001147.
%K nonn,tabf
%O 0,5
%A _Emeric Deutsch_, Nov 05 2006
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