A060170
Number of orbits of length n under the map whose periodic points are counted by A005809.
Original entry on oeis.org
3, 6, 27, 120, 600, 3078, 16611, 91872, 520749, 3004200, 17594247, 104304888, 624801957, 3775722342, 22991161500, 140928011136, 868886416866, 5384796881850, 33525472069563, 209592223788000, 1315211209630794, 8281053081282894, 52301607644921259, 331260902534858976, 2103541885645955625, 13389670112374830378
Offset: 1
a(3) = 27 since a map whose periodic points are counted by A005809 has 3 fixed points and 84 points of period 3, hence 27 orbits of length 3.
Cf.
A005809,
A060164,
A060165,
A060166,
A060167,
A060168,
A060179,
A060171,
A060171,
A060172,
A060173.
-
a(n) = sumdiv(n, d, moebius(n/d)*binomial(3*d, d))/n; \\ Michel Marcus, Sep 10 2017
A256553
Triangle T(n,k) in which the n-th row contains the increasing list of distinct orders of degree-n permutations; n>=0, 1<=k<=A009490(n).
Original entry on oeis.org
1, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 10, 12, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 15, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 20, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 20, 21, 30
Offset: 0
Triangle T(n,k) begins:
1;
1;
1, 2;
1, 2, 3;
1, 2, 3, 4;
1, 2, 3, 4, 5, 6;
1, 2, 3, 4, 5, 6;
1, 2, 3, 4, 5, 6, 7, 10, 12;
1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 15;
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 20;
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 20, 21, 30;
Last elements of rows give
A000793.
-
b:= proc(n, i) option remember; `if`(n=0 or i=1, x,
b(n, i-1)+(p-> add(coeff(p, x, t)*x^ilcm(t, i),
t=1..degree(p)))(add(b(n-i*j, i-1), j=1..n/i)))
end:
T:= n->(p->seq((h->`if`(h=0, [][], i))(coeff(p, x, i))
, i=1..degree(p)))(b(n$2)):
seq(T(n), n=0..12);
-
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x,
b[n, i - 1] + Function[p, Sum[Coefficient[p, x, t]*x^LCM[t, i],
{t, 1, Exponent[p, x]}]][Sum[b[n - i*j, i - 1], {j, 1, n/i}]]];
T[n_] := Function[p, Table[Function[h, If[h == 0, Nothing, i]][
Coefficient[p, x, i]], {i, 1, Exponent[p, x]}]][b[n, n]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jul 15 2021, after Alois P. Heinz *)
A060180
Sum of distinct orders of degree-n even permutations.
Original entry on oeis.org
1, 1, 4, 6, 11, 15, 28, 43, 74, 103, 148, 213, 296, 476, 679, 990, 1133, 1707, 2225, 3260, 4591, 6042, 7343, 9374, 13774, 18262, 25244, 30379, 39768, 47295, 66471, 87903, 115570, 139802, 173605, 215878, 271434, 369256, 466904, 569623, 664775
Offset: 1
Set of orders of all degree 5 even permutations is {1,2,3,5} so a(5)=1+2+3+5=11.
Showing 1-3 of 3 results.
Comments