A124322 - OEIS

%I #19 Jul 23 2024 08:39:22

%S 1,1,1,1,2,3,5,7,3,12,25,15,37,91,60,15,128,329,315,105,457,1415,1533,

%T 630,105,1872,6297,7623,4410,945,8169,29431,42150,27405,7875,945,

%U 37600,151085,233475,176715,69300,10395,188685,802099,1365243,1199220,533610

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

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

%H Alois P. Heinz, <a href="/A124322/b124322.txt">Rows n = 0..200, flattened</a>

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

%e 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 Triangle starts:

%e    1;

%e    1;

%e    1,  1;

%e    2,  3;

%e    5,  7,  3;

%e   12, 25, 15;

%e   37, 91, 60, 15;

%e   ...

%p 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 # 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)=0, 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[Series[Exp[y (Cosh[x] - 1) + Sinh[x]], {x, 0, nn}], {x, y}] // Grid  (* _Geoffrey Critzer_, Aug 28 2012*)

%Y Cf. A000110, A102287, A003724, A124321.

%K nonn,tabf

%O 0,5

%A _Emeric Deutsch_, Oct 28 2006