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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)).

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