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A124322 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 even size (0<=k<=floor(n/2)).

%I A124322 #19 Jul 23 2024 08:39:22
%S A124322 1,1,1,1,2,3,5,7,3,12,25,15,37,91,60,15,128,329,315,105,457,1415,1533,
%T A124322 630,105,1872,6297,7623,4410,945,8169,29431,42150,27405,7875,945,
%U A124322 37600,151085,233475,176715,69300,10395,188685,802099,1365243,1199220,533610
%N A124322 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 even size (0<=k<=floor(n/2)).
%C A124322 Row n has 1+floor(n/2) terms. Sum of row n is the Bell number B(n)=A000110(n). Sum_{k=0..floor(n/2)} k*T(n,k) = A102287(n). T(n,0)=A003724(n).
%D A124322 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 225.
%H A124322 Alois P. Heinz, <a href="/A124322/b124322.txt">Rows n = 0..200, flattened</a>
%F A124322 E.g.f.: exp[sinh(z)+t(cosh(z)-1)].
%e A124322 T(4,1) = 7 because we have 1234, 14|2|3, 1|24|3, 1|2|34, 13|2|4, 1|23|4 and 12|3|4.
%e A124322 Triangle starts:
%e A124322    1;
%e A124322    1;
%e A124322    1,  1;
%e A124322    2,  3;
%e A124322    5,  7,  3;
%e A124322   12, 25, 15;
%e A124322   37, 91, 60, 15;
%e A124322   ...
%p A124322 G:=exp(sinh(z)+t*(cosh(z)-1)): 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,j),j=0..floor(n/2)) od; # yields sequence in triangular form
%p A124322 # second Maple program:
%p A124322 with(combinat):
%p A124322 b:= proc(n, i) option remember; expand(`if`(n=0, 1,
%p A124322       `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
%p A124322       b(n-i*j, i-1)*`if`(irem(i, 2)=0, x^j, 1), j=0..n/i))))
%p A124322     end:
%p A124322 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
%p A124322 seq(T(n), n=0..15);  # _Alois P. Heinz_, Mar 08 2015
%t A124322 nn = 10; Range[0, nn]! CoefficientList[Series[Exp[y (Cosh[x] - 1) + Sinh[x]], {x, 0, nn}], {x, y}] // Grid  (* _Geoffrey Critzer_, Aug 28 2012*)
%Y A124322 Cf. A000110, A102287, A003724, A124321.
%K A124322 nonn,tabf
%O A124322 0,5
%A A124322 _Emeric Deutsch_, Oct 28 2006