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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.
10
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
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
Main diagonal gives A261431.
Cf. A008307.
Sequence in context: A097946 A083926 A218757 * A024466 A021817 A167613
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Aug 18 2015
STATUS
approved