A057740
Irregular triangle read by rows: T(n,k) is the number of elements of alternating group A_n having order k, for n >= 1, 1 <= k <= A051593(n).
Original entry on oeis.org
1, 1, 1, 0, 2, 1, 3, 8, 1, 15, 20, 0, 24, 1, 45, 80, 90, 144, 1, 105, 350, 630, 504, 210, 720, 1, 315, 1232, 3780, 1344, 5040, 5760, 0, 0, 0, 0, 0, 0, 0, 2688, 1, 1323, 5768, 18900, 3024, 37800, 25920, 0, 40320, 9072, 0, 15120, 0, 0, 24192
Offset: 1
Triangle begins:
1;
1;
1, 0, 2;
1, 3, 8;
1, 15, 20, 0, 24;
1, 45, 80, 90, 144;
1, 105, 350, 630, 504, 210, 720;
1, 315, 1232, 3780, 1344, 5040, 5760, 0, 0, 0, 0, 0, 0, 0, 2688;
1, 1323, 5768, 18900, 3024, 37800, 25920, 0, 40320, 9072, 0, 15120, 0, 0, 24192;
...
- J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].
See also
A061129,
A061130,
A061131,
A061132,
A061133,
A061134,
A061135,
A061136,
A061137,
A061138,
A061139,
A061140.
-
{* Order(g) : g in Alt(6) *};
-
row[n_] := (orders = PermutationOrder /@ GroupElements[AlternatingGroup[n] ]; Table[Count[orders, k], {k, 1, Max[orders]}]); Table[row[n], {n, 1, 9}] // Flatten (* Jean-François Alcover, Aug 31 2016 *)
A061137
Number of degree-n odd permutations of order dividing 6.
Original entry on oeis.org
0, 0, 1, 3, 6, 30, 270, 1386, 6048, 46656, 387180, 2469060, 17204616, 158065128, 1903506696, 18887563800, 163657221120, 2095170230016, 30792968596368, 346564643468976, 3905503235814240, 58609511127871200, 866032039742528736
Offset: 0
Cf.
A000085,
A001470,
A001472,
A052501,
A053496-
A053505,
A001189,
A001471,
A001473,
A061121-
A061128,
A000704,
A061129-
A061132,
A048099,
A051695,
A061133-
A061135,
A001465,
A061136-
A061140.
-
m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^3/3)*Sinh(x^2/2 + x^6/6) )); [0,0] cat [Factorial(n+1)*b[n]: n in [1..m-2]]; // G. C. Greubel, Jul 02 2019
-
Egf:= exp(x + x^3/3)*sinh(x^2/2 + x^6/6):
S:= series(Egf,x,31):
seq(coeff(S,x,j)*j!,j=0..30); # Robert Israel, Jul 13 2018
-
With[{m=30}, CoefficientList[Series[Exp[x + x^3/3]*Sinh[x^2/2 + x^6/6], {x, 0, m}], x]*Range[0,m]!] (* Vincenzo Librandi, Jul 02 2019 *)
-
my(x='x+O('x^30)); concat([0,0], Vec(serlaplace( exp(x + x^3/3)*sinh(x^2/2 + x^6/6) ))) \\ G. C. Greubel, Jul 02 2019
-
m = 30; T = taylor(exp(x + x^3/3)*sinh(x^2/2 + x^6/6), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Jul 02 2019
A061138
Number of degree-n odd permutations of order exactly 4.
Original entry on oeis.org
0, 0, 0, 0, 6, 30, 90, 210, 1680, 12096, 114660, 833580, 5928120, 38112360, 259194936, 1739195640, 17043237120, 167089937280, 1837707369840, 18342985021776, 181206905922720, 1673742164139360, 16992525855006240
Offset: 0
Cf.
A000085,
A001470,
A001472,
A052501,
A053496 -
A053505,
A001189,
A001471,
A001473,
A061121 -
A061128,
A000704,
A061129 -
A061132,
A048099,
A051695,
A061133 -
A061135,
A001465,
A061136 -
A061140.
A061139
Number of degree-n odd permutations of order exactly 6.
Original entry on oeis.org
0, 0, 0, 0, 0, 20, 240, 1260, 5600, 45360, 383040, 2451680, 17128320, 157769040, 1902380480, 18882623760, 163633317120, 2095059774080, 30792478993920, 346562329685760, 3905491275514880, 58609449249207360, 866031730098205440
Offset: 0
Cf.
A000085,
A001470,
A001472,
A052501,
A053496 -
A053505,
A001189,
A001471,
A001473,
A061121 -
A061128,
A000704,
A061129 -
A061132,
A048099,
A051695,
A061133 -
A061135,
A001465,
A061136 -
A061140.
A308648
Number of degree-n odd permutations of order dividing 8.
Original entry on oeis.org
0, 0, 1, 3, 12, 40, 120, 336, 7168, 58752, 345600, 1682560, 15983616, 142192128, 2318697472, 25614382080, 282753361920, 2645093410816, 48869743454208, 674729909839872, 12153962014842880, 167314499427532800, 1986101341059956736, 20335611320801886208
Offset: 0
For n=3 the a(3)=3 solutions are (1, 2), (1, 3), (2, 3) (permutations in cyclic notation).
-
With[{nn = 22},
CoefficientList[Series[1/2 Exp[x + x^2/2 + x^4/4 + x^8/8] - 1/2 Exp[x - x^2/2 - x^4/4 - x^8/8], {x, 0, nn}], x]*Range[0, nn]!]
Showing 1-5 of 5 results.