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A124321 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).

%I A124321 #17 Sep 05 2025 04:42:05
%S A124321 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 A124321 0,441,0,56,0,1,379,0,2500,0,1176,0,84,0,1,0,6556,0,11740,0,2730,0,
%U A124321 120,0,1,6556,0,59671,0,43870,0,5712,0,165,0,1,0,150349,0,378356,0,138622,0
%N A124321 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 A124321 Row sums are the Bell numbers (A000110).
%C A124321 Sum_{k=0..n} k*T(n,k) = A102286(n).
%C A124321 T(2*n,0) = A005046(n); T(2*n+1,0) = 0.
%D A124321 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 225.
%H A124321 Alois P. Heinz, <a href="/A124321/b124321.txt">Rows n = 0..200, flattened</a>
%F A124321 E.g.f.: G(t,z) = exp(t*sinh(z)+cosh(z)-1).
%e A124321 T(3,1) = 4 because we have 123, 1|23, 12|3 and 13|2.
%e A124321 Triangle starts:
%e A124321   1;
%e A124321   0,  1;
%e A124321   1,  0,  1;
%e A124321   0,  4,  0,  1;
%e A124321   4,  0, 10,  0,  1;
%e A124321   0, 31,  0, 20,  0,  1;
%p A124321 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 A124321 # second Maple program:
%p A124321 with(combinat):
%p A124321 b:= proc(n, i) option remember; expand(`if`(n=0, 1,
%p A124321       `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
%p A124321       b(n-i*j, i-1)*`if`(irem(i, 2)=1, x^j, 1), j=0..n/i))))
%p A124321     end:
%p A124321 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
%p A124321 seq(T(n), n=0..15);  # _Alois P. Heinz_, Mar 08 2015
%t A124321 nn = 10; Range[0, nn]! CoefficientList[Series[Exp[ (Cosh[x] - 1) + y Sinh[x]], {x, 0, nn}], {x, y}] // Grid (* _Geoffrey Critzer_, Aug 28 2012 *)
%Y A124321 Cf. A000110, A102286, A005046, A124322.
%K A124321 nonn,tabl,changed
%O A124321 0,8
%A A124321 _Emeric Deutsch_, Oct 28 2006