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

Author

Emeric Deutsch, Oct 28 2006

Keywords

• nonn
• tabl
• changed

Comments

Row sums are the Bell numbers (A000110).

Sum_{k=0..n} k*T(n,k) = A102286(n).

T(2*n,0) = A005046(n); T(2*n+1,0) = 0.

Examples

			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;
		

References

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

Links

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

Crossrefs

Cf. A000110, A102286, A005046, A124322.

Programs

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 *)

Formula

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