A261430 Number A(n,k) of permutations p of [n] without fixed points such that p^k = Id; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 2, 3, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 9, 0, 15, 0, 0, 1, 0, 0, 2, 0, 0, 40, 0, 0, 0, 1, 0, 1, 0, 3, 24, 105, 0, 105, 0, 0, 1, 0, 0, 0, 0, 20, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 9, 0, 175, 0, 2625, 2240, 945, 0, 0

Offset

0,25

Links

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

Formula

E.g.f. of column k: exp(Sum_{d|k, d>1} x^d/d).

Example

Square array A(n,k) begins:
  1, 1,   1,  1,    1,  1,    1,   1,    1, ...
  0, 0,   0,  0,    0,  0,    0,   0,    0, ...
  0, 0,   1,  0,    1,  0,    1,   0,    1, ...
  0, 0,   0,  2,    0,  0,    2,   0,    0, ...
  0, 0,   3,  0,    9,  0,    3,   0,    9, ...
  0, 0,   0,  0,    0, 24,   20,   0,    0, ...
  0, 0,  15, 40,  105,  0,  175,   0,  105, ...
  0, 0,   0,  0,    0,  0,  210, 720,    0, ...
  0, 0, 105,  0, 2625,  0, 4585,   0, 7665, ...

Maple

with(numtheory):
A:= proc(n, k) option remember; `if`(n<0, 0, `if`(n=0, 1,
      add(mul(n-i, i=1..j-1)*A(n-j, k), j=divisors(k) minus {1})))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);

Mathematica

A[0, 0] = A[0, 1] = 1; A[_, 0|1] = 0; A[n_, k_] := A[n, k] = If[n < 0, 0, If[n == 0, 1, Sum[Product[n - i, {i, 1, j - 1}]*A[n - j, k], {j, Rest @ Divisors[k]}]]]; Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jan 21 2017, after Alois P. Heinz *)

Crossrefs

Columns k=0+1,2-10 give: A000007, A001147, A052502, A052503, A052504, A261317, A261427, A261428, A261429, A261381.

Main diagonal gives A261431.

Cf. A008307.

Sequence in context: A097946 A083926 A218757 * A024466 A021817 A069585

Adjacent sequences: A261427 A261428 A261429 * A261431 A261432 A261433

Keyword

nonn,tabl

Author

Alois P. Heinz, Aug 18 2015

Status

approved