A053195 - OEIS

%I #18 Jul 07 2015 04:29:08

%S 1,1,2,3,18,45,360,1575,20790,99225,1332450,9823275,181496700,

%T 1404728325,26221595400,273922023375,7196040101250,69850115960625,

%U 1662139682453250,22561587455281875,675158520854317500,9002073394657468125,259715927440434465000

%N Number of level permutations of degree n.

%C A permutation is level if the powers of 2 dividing its cycle lengths are all equal.

%C For odd n, level permutations of degree n are just permutations that have odd order, i.e., A053195(2*n+1) = A000246(2*n+1). - _Vladeta Jovovic_, Sep 29 2004

%H Alois P. Heinz, <a href="/A053195/b053195.txt">Table of n, a(n) for n = 0..400</a>

%H L. Babai and P. J. Cameron, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v7i1r38/0">Automorphisms and enumeration of switching classes of tournaments</a>, Electron. J. Combin., 7 (2000), no. 1, Research Paper 38, 25 pp.

%p with(combinat):

%p b:= proc(n, i, p) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p        add(multinomial(n, n-i*j, i$j)/j!*(i-1)!^j*

%p        b(n-i*j, i-2*p, p), j=0..n/i)))

%p     end:

%p a:= n-> (m-> `if`(n=0, 1, add(b(n, (h-> h-1+irem(h, 2)

%p     )(iquo(n, 2^j))*2^j, 2^j), j=0..m)))(ilog2(n)):

%p seq(a(n), n=0..25);  # _Alois P. Heinz_, Jun 11 2015

%t multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_, p_] := b[n, i, p] = If[n == 0, 1, If[i<1, 0, Sum[multinomial[n, Join[{n-i*j}, Array[i&, j]]]/j!*(i-1)!^j*b[n - i*j, i-2*p, p], {j, 0, n/i}]]]; a[n_] := Function[{m}, If[n == 0, 1, Sum[ b[n, Function [{h}, h - 1 + Mod[h, 2]][Quotient[n, 2^j]]*2^j, 2^j], {j, 0, m}]]][Log[2, n] // Floor]; Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Jul 07 2015, after _Alois P. Heinz_ *)

%Y Cf. A049313, A053197.

%K nonn,nice

%O 0,3

%A _Vladeta Jovovic_, Mar 02 2000

%E a(0)=1 prepended by _Alois P. Heinz_, Jun 11 2015