A326241
Number of degree-n even permutations of order dividing 12.
Original entry on oeis.org
1, 1, 1, 3, 12, 36, 216, 1296, 10368, 78912, 634896, 5572656, 51817536, 477672768, 8268884352, 101752505856, 1417554660096, 20985416983296, 344834432195328, 5096129755468032, 70148917686998016
Offset: 0
For n=3 the a(3)=3 solutions are (1), (1, 2, 3), (1, 3, 2) (permutations in cyclic notation).
- J. Riordan, An Introduction to Combinatorial Analysis, John Wiley & Sons, Inc. New York, 1958 (Chap 4, Problem 22).
Cf.
A053502,
A326242,
A000704,
A061130,
A061131,
A061132,
A048099,
A051695,
A061133,
A061134,
A061135,
A326242.
-
E:= (1/2)*exp(x + (1/2)*x^2 + (1/3)*x^3 + (1/4)*x^4 + (1/6)*x^6+(1/12)*x^(12)) + (1/2)*exp(x - (1/2)*x^2 + (1/3)*x^3 - (1/4)*x^4 - (1/6)*x^6-(1/12)*x^(12)):
S:= series(E,x,31):
seq(coeff(S,x,i)*i!,i=0..30);# Robert Israel, Jul 08 2019
-
With[{nn = 22}, CoefficientList[Series[1/2 Exp[x + x^2/2 + x^3/3 + x^4/4 + x^6/6 +x^12/12]+1/2 Exp[x - x^2/2 + x^3/3 - x^4/4 - x^6/6 - x^12/12], {x, 0, nn}], x]*Range[0, nn]!]
A326242
Number of degree-n odd permutations of order dividing 12.
Original entry on oeis.org
0, 0, 1, 3, 12, 60, 360, 2016, 11088, 73872, 602640, 4411440, 81677376, 934435008, 8100473472, 104370819840, 1448725616640, 15823660179456, 247231858514688, 3703908371910912, 66727356304757760, 1124506454958351360, 19305439846610835456
Offset: 0
For n=3 the a(3)=3 solutions are (1, 2), (2, 3), (1, 3) (permutations in cyclic notation).
Cf.
A053502,
A326242,
A000704,
A061130,
A061131,
A061132,
A048099,
A051695,
A061133,
A061134,
A061135,
A326241.
-
E:= (1/2)*exp(x + (1/2)*x^2 + (1/3)*x^3 + (1/4)*x^4 + (1/6)*x^6+(1/12)*x^(12)) - (1/2)*exp(x - (1/2)*x^2 + (1/3)*x^3 - (1/4)*x^4 - (1/6)*x^6-(1/12)*x^(12)):
S:= series(E,x,31):
seq(coeff(S,x,i)*i!,i=0..30); # Robert Israel, Jul 08 2019
-
With[{nn = 22}, CoefficientList[Series[1/2 Exp[x + x^2/2 + x^3/3 + x^4/4 + x^6/6 +x^12/12]-1/2 Exp[x - x^2/2 + x^3/3 - x^4/4 - x^6/6 - x^12/12], {x, 0, nn}], x]*Range[0, nn]!]
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.