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).

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, 0, 441, 0, 56, 0, 1, 379, 0, 2500, 0, 1176, 0, 84, 0, 1, 0, 6556, 0, 11740, 0, 2730, 0, 120, 0, 1, 6556, 0, 59671, 0, 43870, 0, 5712, 0, 165, 0, 1, 0, 150349, 0, 378356, 0, 138622, 0

Offset

0,8

Comments

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.

References

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

Links

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

Formula

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

Example

T(3,1)=4 because we have 123, 1|23, 12|3 and 13|2.
Triangle starts:
1;
0,1;
1,0,1;
0,4,0,1;
4,0,10,0,1;
0,31,0,20,0,1;

Maple

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

Crossrefs

Cf. A000110, A102286, A005046, A124322.

Sequence in context: A213600 A178104 A172545 * A232195 A309333 A298924

Adjacent sequences: A124318 A124319 A124320 * A124322 A124323 A124324

Keyword

nonn,tabl

Author

Emeric Deutsch, Oct 28 2006

Status

approved