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Triangle read by rows: T(n,k) is the number of set partitions of {1,2,...,n} (or of any n-set) having k blocks of odd size (0<=k<=n).
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%I #11 Mar 08 2015 04:36:23

%S 1,0,1,1,0,1,0,4,0,1,4,0,10,0,1,0,31,0,20,0,1,31,0,136,0,35,0,1,0,379,

%T 0,441,0,56,0,1,379,0,2500,0,1176,0,84,0,1,0,6556,0,11740,0,2730,0,

%U 120,0,1,6556,0,59671,0,43870,0,5712,0,165,0,1,0,150349,0,378356,0,138622,0

%N Triangle read by rows: T(n,k) is the number of set partitions of {1,2,...,n} (or of any n-set) having k blocks of odd size (0<=k<=n).

%C Row sums are the Bell numbers (A000110). Sum(k*T(n,k),k=0..n)=A102286(n). T(2n,0)=A005046(n); T(2n+1,0)=0.

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 225.

%H Alois P. Heinz, <a href="/A124321/b124321.txt">Rows n = 0..200, flattened</a>

%F E.g.f.: G(t,z)=exp[t*sinh(z)+cosh(z)-1].

%e T(3,1)=4 because we have 123, 1|23, 12|3 and 13|2.

%e Triangle starts:

%e 1;

%e 0,1;

%e 1,0,1;

%e 0,4,0,1;

%e 4,0,10,0,1;

%e 0,31,0,20,0,1;

%p G:=exp(t*sinh(z)+cosh(z)-1): Gser:=simplify(series(G,z=0,15)): for n from 0 to 12 do P[n]:=sort(n!*coeff(Gser,z,n)) od: for n from 0 to 12 do seq(coeff(P[n],t,j),j=0..n) 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`(irem(i, 2)=1, 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 nn = 10; Range[0, nn]! CoefficientList[

%t Series[Exp[ (Cosh[x] - 1) + y Sinh[x]], {x, 0, nn}], {x, y}] // Grid (* Geoffrey Critzer, Aug 28 2012 *)

%Y Cf. A000110, A102286, A005046, A124322.

%K nonn,tabl

%O 0,8

%A _Emeric Deutsch_, Oct 28 2006