A237996 Triangular array read by rows. T(n,k) is the number of even permutations of {1,2,...,n} that have exactly k cycles, n>=0,0<=k<=n.

1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 11, 0, 1, 0, 24, 0, 35, 0, 1, 0, 0, 274, 0, 85, 0, 1, 0, 720, 0, 1624, 0, 175, 0, 1, 0, 0, 13068, 0, 6769, 0, 322, 0, 1, 0, 40320, 0, 118124, 0, 22449, 0, 546, 0, 1, 0, 0, 1026576, 0, 723680, 0, 63273, 0, 870, 0, 1

Offset

0,8

Comments

Row sums = A001710(n).

References

J. Riordan, Introduction to Combinatorial Analysis, Wiley, 1958, page 87, problem # 20.

Links

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

Formula

E.g.f.: exp(y*A(x))*cosh(y*B(x)) where A(x)= log((1 + x)/(1 - x))^(1/2) and B(x)=log(1/(1-x^2)^(1/2).

Example

1,
0, 1,
0, 0, 1,
0, 2, 0, 1,
0, 0, 11, 0, 1,
0, 24, 0, 35, 0, 1,
0, 0, 274, 0, 85, 0, 1,
0, 720, 0, 1624, 0, 175, 0, 1,
0, 0, 13068, 0, 6769, 0, 322, 0, 1,
0, 40320, 0, 118124, 0, 22449, 0, 546, 0, 1,
0, 0, 1026576, 0, 723680, 0, 63273, 0, 870, 0, 1

Maple

with(combinat):
b:= proc(n, i, t) option remember; expand(`if`(n=0, t, `if`(i<1,
      0, add(x^j*multinomial(n, n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j,
      i-1, irem(t+`if`(irem(i, 2)=0, j, 0), 2)), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 1)):
seq(T(n), n=0..12);  # Alois P. Heinz, Mar 09 2015

Mathematica

nn=11; a=Log[((1+x)/(1-x))^(1/2)]; b=Log[1/(1-x^2)^(1/2)]; Table[Take[(Range[0, nn]!CoefficientList[Series[Exp[y a]Cosh[y b] , {x, 0, nn}], {x, y}])[[n]], n], {n, 1, nn}]//Grid

Crossrefs

Sequence in context: A111594 A322549 A349645 * A203951 A323591 A105348

Adjacent sequences: A237993 A237994 A237995 * A237997 A237998 A237999

Keyword

nonn,tabl

Author

Geoffrey Critzer, Feb 16 2014

Status

approved