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)).

1, 1, 1, 1, 2, 3, 5, 7, 3, 12, 25, 15, 37, 91, 60, 15, 128, 329, 315, 105, 457, 1415, 1533, 630, 105, 1872, 6297, 7623, 4410, 945, 8169, 29431, 42150, 27405, 7875, 945, 37600, 151085, 233475, 176715, 69300, 10395, 188685, 802099, 1365243, 1199220, 533610

Offset

0,5

Comments

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).

References

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

Links

Alois P. Heinz, Rows n = 0..200, flattened

Formula

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

Example

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.
Triangle starts:
   1;
   1;
   1,  1;
   2,  3;
   5,  7,  3;
  12, 25, 15;
  37, 91, 60, 15;
  ...

Maple

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
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
      b(n-i*j, i-1)*`if`(irem(i, 2)=0, x^j, 1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..15);  # Alois P. Heinz, Mar 08 2015

Mathematica

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*)

Crossrefs

Cf. A000110, A102287, A003724, A124321.

Sequence in context: A163821 A156294 A126607 * A209037 A334455 A075241

Adjacent sequences: A124319 A124320 A124321 * A124323 A124324 A124325

Keyword

nonn,tabf

Author

Emeric Deutsch, Oct 28 2006

Status

approved