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A213279 Triangle read by rows: T(n,k) (n>=1, 1 <= k <= n) = number of permutations of [1..n] in which all cycle lengths are divisible by k. 2
1, 2, 1, 6, 0, 2, 24, 9, 0, 6, 120, 0, 0, 0, 24, 720, 225, 160, 0, 0, 120, 5040, 0, 0, 0, 0, 0, 720, 40320, 11025, 0, 6300, 0, 0, 0, 5040, 362880, 0, 62720, 0, 0, 0, 0, 0, 40320, 3628800, 893025, 0, 0, 435456, 0, 0, 0, 0, 362880, 39916800, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3628800 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Alois P. Heinz, Rows n = 1..141, flattened

E. D. Bolker and A. M. Gleason, Counting permutations, J. Combin. Thy., A 29 (1980), 236-242.

EXAMPLE

Triangle begins

[1],

[2, 1],

[6, 0, 2],

[24, 9, 0, 6],

[120, 0, 0, 0, 24],

[720, 225, 160, 0, 0, 120],

[5040, 0, 0, 0, 0, 0, 720],

[40320, 11025, 0, 6300, 0, 0, 0, 5040],

[362880, 0, 62720, 0, 0, 0, 0, 0, 40320],

[3628800, 893025, 0, 0, 435456, 0, 0, 0, 0, 362880],

[39916800, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3628800],

[479001600, 108056025, 68992000, 56133000, 0, 46569600, 0, 0, 0, 0, 0, 39916800],

...

MAPLE

read transforms;

f:=(n, d)->mul(n-j+did(j, d), j=1..n); # did(d, j) = 1 iff j divides d, otherwise 0

g:=n->[seq(f(n, d), d=1..n)];

[seq(g(n), n=1..14)];

# second Maple program:

T:= proc(n, k) option remember; `if`(n=0, 1, add(

      `if`(irem(j, k)=0, binomial(n-1, j-1)*(j-1)!*

       T(n-j, k), 0), j=1..n))

    end:

seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, May 14 2016

MATHEMATICA

T[n_, k_] := T[n, k] = If[n == 0, 1, Sum[If[Mod[j, k] == 0, Binomial[n-1, j -1] * (j-1)! * T[n-j, k], 0], {j, 1, n}]];

Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-Fran├žois Alcover, May 24 2016, after Alois P. Heinz *)

CROSSREFS

Cf. A213280.

Sequence in context: A110863 A215219 A091615 * A175591 A187784 A198812

Adjacent sequences:  A213276 A213277 A213278 * A213280 A213281 A213282

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Jun 08 2012

STATUS

approved

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Last modified March 28 07:59 EDT 2020. Contains 333079 sequences. (Running on oeis4.)