A060523 Triangle T(n,k) = number of degree-n permutations with k even cycles, k=0..n.

1, 1, 0, 1, 1, 0, 3, 3, 0, 0, 9, 12, 3, 0, 0, 45, 60, 15, 0, 0, 0, 225, 345, 135, 15, 0, 0, 0, 1575, 2415, 945, 105, 0, 0, 0, 0, 11025, 18480, 9030, 1680, 105, 0, 0, 0, 0, 99225, 166320, 81270, 15120, 945, 0, 0, 0, 0, 0, 893025, 1596105, 897750, 217350, 23625, 945, 0, 0, 0, 0, 0

Offset: 0

Author

Vladeta Jovovic, Apr 01 2001

Keywords

• easy
• nonn
• tabl

Examples

			Triangle T(n,k) begins:
       1;
       1,       0;
       1,       1,      0;
       3,       3,      0,      0;
       9,      12,      3,      0,     0;
      45,      60,     15,      0,     0,   0;
     225,     345,    135,     15,     0,   0, 0;
    1575,    2415,    945,    105,     0,   0, 0, 0;
   11025,   18480,   9030,   1680,   105,   0, 0, 0, 0;
   99225,  166320,  81270,  15120,   945,   0, 0, 0, 0, 0;
  893025, 1596105, 897750, 217350, 23625, 945, 0, 0, 0, 0, 0;
  ...
		

References

• I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, p. 189, Exercise 3.3.13.

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Columns k=0-1 give: A000246, A096471.

Row sums give A000142.

Cf. A060524, A092691.

Programs

Maple

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)*(i-1)!^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..n))(b(n$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Mar 09 2015

Mathematica

nn = 6; Range[0, nn]! CoefficientList[
   Series[(1 - x^2)^(-y/2) ((1 + x)/(1 - x))^(1/2), {x, 0, nn}], {x, y}] // Grid  (* Geoffrey Critzer, Aug 27 2012 *)

Formula

E.g.f.: (1+x)^((1-y)/2)/(1-x)^((1+y)/2).

Sum_{k=0..n} k * T(n,k) = A092691(n). - Alois P. Heinz, Aug 17 2023