A053505 -id:A053505 - cp's OEIS frontend

A061135 Number of degree-n even permutations of order exactly 10.

0, 0, 0, 0, 0, 0, 0, 0, 9072, 90720, 498960, 25945920, 321080760, 2460970512, 14552417880, 115251776640, 4603779180000, 72193873752000, 681167139805152, 16976210865344640, 304992335584165320, 4548189212204243760

Offset: 1

Vladeta Jovovic, Apr 14 2001

Cf. A000085, A001470, A001472, A052501, A053496 - A053505, A001189, A001471, A001473, A061121 - A061128, A000704, A061129 - A061132, A048099, A051695, A061133 - A061135.

E.g.f.: exp(x) - 1/2*exp(x + 1/2*x^2) - 1/2*exp(x - 1/2*x^2) - exp(x + 1/5*x^5) + 1/2*exp(x + 1/2*x^2 + 1/5*x^5 + 1/10*x^10) + 1/2*exp(x - 1/2*x^2 + 1/5*x^5 - 1/10*x^10).

A061129 Number of degree-n even permutations of order dividing 4.

Original entry on oeis.org

1, 1, 1, 1, 4, 16, 136, 736, 4096, 20224, 99856, 475696, 3889216, 31778176, 313696384, 2709911296, 23006784256, 179965340416, 1532217039616, 13081112406784, 147235213351936, 1657791879049216, 20132199908571136, 226466449808367616, 2542933338768769024

Offset: 0

Vladeta Jovovic, Apr 14 2001

Alois P. Heinz, Table of n, a(n) for n = 0..570
Lev Glebsky, Melany Licón, Luis Manuel Rivera, On the number of even roots of permutations, arXiv:1907.00548 [math.CO], 2019.

Cf. A000085, A001470, A001472, A052501, A053496 - A053505, A001189, A001471, A001473, A061121 - A061128, A000704, A061129 - A061132, A048099, A051695, A061133 - A061135.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x)*Cosh(x^2/2 + x^4/4) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jul 02 2019

With[{n=30}, CoefficientList[Series[Exp[x]*Cosh[x^2/2 + x^4/4], {x, 0, n}], x]*Range[0, n]!] (* G. C. Greubel, Jul 02 2019 *)

my(x='x+O('x^30)); Vec(serlaplace( exp(x)*cosh(x^2/2 + x^4/4) )) \\ G. C. Greubel, Jul 02 2019

m = 30; T = taylor(exp(x)*cosh(x^2/2 + x^4/4), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Jul 02 2019

E.g.f.: exp(x)*cosh(x^2/2 + x^4/4).

A053497 Number of degree-n permutations of order dividing 7.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 721, 5761, 25921, 86401, 237601, 570241, 1235521, 892045441, 13348249201, 106757164801, 604924594561, 2722120577281, 10344007402561, 34479959558401, 24928970490633601, 546446134633639681, 6281586217487489041, 50248618811434961281

Offset: 0

N. J. A. Sloane, Jan 15 2000

nonn

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.

Alois P. Heinz, Table of n, a(n) for n = 0..200
L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.

Sequences with e.g.f. exp(x + x^m/m): A000079 (m=1), A000085 (m=2), A001470 (m=3), A118934 (m=4), A052501 (m=5), A293588 (m=6), this sequence (m=7).
Cf. A000085, A001470, A001472, A005388, A053495 - A053505, A261427.
Column k=7 of A008307.

R:=PowerSeriesRing(Rationals(), 31); Coefficients(R!(Laplace( Exp(x + x^7/7) ))); // G. C. Greubel, May 14 2019, Mar 07 2021

a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
       add(mul(n-i, i=1..j-1)*a(n-j), j=[1, 7])))
    end:
seq(a(n), n=0..25); # Alois P. Heinz, Feb 14 2013

CoefficientList[Series[Exp[x+x^7/7], {x, 0, 24}], x]*Range[0, 24]! (* Jean-François Alcover, Mar 24 2014 *)

my(x='x+O('x^30)); Vec(serlaplace( exp(x+x^7/7) )) \\ G. C. Greubel, May 14 2019

f=factorial; [sum(f(n)/(7^j*f(j)*f(n-7*j)) for j in (0..n/7)) for n in (0..30)] # G. C. Greubel, May 14 2019

E.g.f.: exp(x + x^7/7).

a(n) = Sum_{k=0..floor(n/7)} n!/(7^k*k!*(n-7*k)!). - G. C. Greubel, Mar 07 2021

A061136 Number of degree-n odd permutations of order dividing 4.

Original entry on oeis.org

0, 0, 1, 3, 12, 40, 120, 336, 2128, 13392, 118800, 850960, 6004416, 38408448, 260321152, 1744135680, 17067141120, 167200393216, 1838196972288, 18345298804992, 181218866222080, 1673804042803200, 16992835499329536

Offset: 0

Vladeta Jovovic, Apr 14 2001

Lev Glebsky, Melany Licón, Luis Manuel Rivera, On the number of even roots of permutations, arXiv:1907.00548 [math.CO], 2019.

Cf. A000085, A001470, A001472, A052501, A053496 - A053505, A001189, A001471, A001473, A061121 - A061128, A000704, A061129 - A061132, A048099, A051695, A061133 - A061135, A001465, A061136 - A061140.

E.g.f.: 1/2*exp(x + 1/2*x^2 + 1/4*x^4) - 1/2*exp(x - 1/2*x^2 - 1/4*x^4).

A053499 Number of degree-n permutations of order dividing 9.

Original entry on oeis.org

1, 1, 1, 3, 9, 21, 81, 351, 1233, 46089, 434241, 2359611, 27387801, 264333213, 1722161169, 16514298711, 163094452641, 1216239520401, 50883607918593, 866931703203699, 8473720481213481, 166915156382509221, 2699805625227141201, 28818706120636531023, 439756550972215638129, 6766483260087819272601, 77096822666547068590401, 3568144263578808757678251

Offset: 0

N. J. A. Sloane, Jan 15 2000

nonn

Differs from A218003 first at n=27. - Alois P. Heinz, Jan 25 2014

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.

Alois P. Heinz, Table of n, a(n) for n = 0..200
L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.

Cf. A000085, A001470, A001472, A053495-A053505, A005388, A261429.

Column k=9 of A008307.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^3/3 + x^9/9) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 15 2019

a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
       add(mul(n-i, i=1..j-1)*a(n-j), j=[1, 3, 9])))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 14 2013

CoefficientList[Series[Exp[x+x^3/3+x^9/9], {x, 0, 30}], x]*Range[0, 30]! (* Jean-François Alcover, Mar 24 2014 *)

my(x='x+O('x^30)); Vec(serlaplace( exp(x + x^3/3 + x^9/9) )) \\ G. C. Greubel, May 15 2019

m = 30; T = taylor(exp(x + x^3/3 + x^9/9), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 15 2019

E.g.f.: exp(x + x^3/3 + x^9/9).

A053502 Number of degree-n permutations of order dividing 12.

Original entry on oeis.org

1, 1, 2, 6, 24, 96, 576, 3312, 21456, 152784, 1237536, 9984096, 133494912, 1412107776, 16369357824, 206123325696, 2866280276736, 36809077162752, 592066290710016, 8800038127378944, 136876273991755776, 2197453620220010496, 37915306084793106432

Offset: 0

N. J. A. Sloane, Jan 15 2000

nonn

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.

Alois P. Heinz, Table of n, a(n) for n = 0..200
L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.

Cf. A000085, A001470, A001472, A053495-A053505, A005388.

Column k=12 of A008307.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^2/2 + x^3/3 + x^4/4 + x^6/6 + x^12/12) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 15 2019

a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
       add(mul(n-i, i=1..j-1)*a(n-j), j=[1, 2, 3, 4, 6, 12])))
    end:
seq(a(n), n=0..25); # Alois P. Heinz, Feb 14 2013

a[n_]:= a[n] = If[n<0, 0, If[n==0, 1, Sum[Product[n-i, {i, 1, j-1}]*a[n-j], {j, {1, 2, 3, 4, 6, 12}}]]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 24 2014, after Alois P. Heinz *)
With[{m = 30}, CoefficientList[Series[Exp[x +x^2/2 +x^3/3 +x^4/4 +x^6/6 + x^12/12], {x, 0, m}], x]*Range[0, m]!] (* G. C. Greubel, May 15 2019 *)

my(x='x+O('x^30)); Vec(serlaplace( exp(x + x^2/2 + x^3/3 + x^4/4 + x^6/6 + x^12/12) )) \\ G. C. Greubel, May 15 2019

m = 30; T = taylor(exp(x + x^2/2 + x^3/3 + x^4/4 + x^6/6 + x^12/12), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 15 2019

E.g.f.: exp(x + x^2/2 + x^3/3 + x^4/4 + x^6/6 + x^12/12).

A061131 Number of degree-n even permutations of order dividing 8.

Original entry on oeis.org

1, 1, 1, 1, 4, 16, 136, 736, 4096, 20224, 326656, 2970496, 33826816, 291237376, 2129910784, 13607197696, 324498374656, 4599593353216, 52741679343616, 495632154179584, 7127212838772736, 94268828128854016, 2098358019107700736, 34030412427789500416

Offset: 0

Vladeta Jovovic, Apr 14 2001

J. Riordan, An Introduction to Combinatorial Analysis, John Wiley & Sons, Inc. New York, 1958 (Chap 4, Problem 22).

Alois P. Heinz, Table of n, a(n) for n = 0..502
Lev Glebsky, Melany Licón, Luis Manuel Rivera, On the number of even roots of permutations, arXiv:1907.00548 [math.CO], 2019.
T. Koda, M. Sato, Y. Tskegahara, 2-adic properties for the numbers of involutions in the alternating groups, J. Algebra Appl. 14 (2015), no. 4, 1550052 (21 pages).

Cf. A000085, A001470, A001472, A052501, A053496-A053505, A001189, A001471, A001473, A061121 - A061128, A000704, A061129-A061132, A048099, A051695, A061133-A061135.

my(x='x+O('x^30)); Vec(serlaplace(1/2*exp(x + 1/2*x^2 + 1/4*x^4 + 1/8*x^8) + 1/2*exp(x - 1/2*x^2 - 1/4*x^4 - 1/8*x^8))) \\ Michel Marcus, Jun 18 2019

E.g.f.: 1/2*exp(x + 1/2*x^2 + 1/4*x^4 + 1/8*x^8) + 1/2*exp(x - 1/2*x^2 - 1/4*x^4 - 1/8*x^8).

A061140 Number of degree-n odd permutations of order exactly 8.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 5040, 45360, 226800, 831600, 9979200, 103783680, 2058376320, 23870246400, 265686220800, 2477893017600, 47031546481920, 656384611034880, 11972743148620800, 165640695384729600, 1969108505560627200

Offset: 0

Vladeta Jovovic, Apr 14 2001

Cf. A000085, A001470, A001472, A052501, A053496 - A053505, A001189, A001471, A001473, A061121 - A061128, A000704, A061129 - A061132, A048099, A051695, A061133 - A061135.

E.g.f.: - 1/2*exp(x + 1/2*x^2 + 1/4*x^4) + 1/2*exp(x - 1/2*x^2 - 1/4*x^4) + 1/2*exp(x + 1/2*x^2 + 1/4*x^4 + 1/8*x^8) - 1/2*exp(x - 1/2*x^2 - 1/4*x^4 - 1/8*x^8).

A053498 Number of degree-n permutations of order dividing 8.

Original entry on oeis.org

1, 1, 2, 4, 16, 56, 256, 1072, 11264, 78976, 672256, 4653056, 49810432, 433429504, 4448608256, 39221579776, 607251736576, 7244686764032, 101611422797824, 1170362064019456, 19281174853615616, 261583327556386816, 4084459360167657472, 54366023748591386624

Offset: 0

N. J. A. Sloane, Jan 15 2000

nonn

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.

Alois P. Heinz, Table of n, a(n) for n = 0..200
L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.

Cf. A000085, A001470, A001472, A053495-A053505, A005388, A261428.

Column k=8 of A008307.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x +x^2/2 +x^4/4 +x^8/8) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 14 2019

a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
       add(mul(n-i, i=1..j-1)*a(n-j), j=[1, 2, 4, 8])))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 14 2013

CoefficientList[Series[Exp[x+x^2/2+x^4/4+x^8/8], {x, 0, 23}], x]*Range[0, 23]! (* Jean-François Alcover, Mar 24 2014 *)

my(x='x+O('x^30)); Vec(serlaplace( exp(x +x^2/2 +x^4/4 +x^8/8) )) \\ G. C. Greubel, May 14 2019

m = 30; T = taylor(exp(x +x^2/2 +x^4/4 +x^8/8), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 14 2019

E.g.f.: exp(x + x^2/2 + x^4/4 + x^8/8).

A053504 Number of degree-n permutations of order dividing 24.

Original entry on oeis.org

1, 1, 2, 6, 24, 96, 576, 3312, 26496, 198144, 1691136, 14973696, 193370112, 2034809856, 25087186944, 313539434496, 4421478721536, 58307347556352, 915011420737536, 13553664911437824, 240637745416421376, 3965015057937924096

Offset: 0

N. J. A. Sloane, Jan 15 2000

nonn

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.

Alois P. Heinz, Table of n, a(n) for n = 0..200
L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.

Cf. A000085, A001470, A001472, A053495-A053505, A005388.

Column k=24 of A008307.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x +x^2/2 +x^3/3 +x^4/4 +x^6/6 +x^8/8 +x^12/12 +x^24/24) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 15 2019

a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
       add(mul(n-i, i=1..j-1)*a(n-j), j=[1, 2, 3, 4, 6, 8, 12, 24])))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 25 2014

a[n_]:= a[n] = If[n<0, 0, If[n==0, 1, Sum[Product[n-i, {i, 1, j-1}]*a[n-j], {j, {1, 2, 3, 4, 6, 8, 12, 24}}]]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 19 2014, after Alois P. Heinz *)
With[{nn=30},CoefficientList[Series[Exp[Total[x^#/#&/@Divisors[24]]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 05 2016 *)

N=30; x='x+O('x^N);
Vec(serlaplace(exp(sumdiv(24, d, x^d/d)))) \\ Gheorghe Coserea, May 11 2017

m = 30; T = taylor(exp(x +x^2/2 +x^3/3 +x^4/4 +x^6/6 +x^8/8 +x^12/12 +x^24/24), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 15 2019

E.g.f.: exp(x + x^2/2 + x^3/3 + x^4/4 + x^6/6 + x^8/8 + x^12/12 + x^24/24).

cp's OEIS Frontend

A061135 Number of degree-n even permutations of order exactly 10.

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A061129 Number of degree-n even permutations of order dividing 4.

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A053497 Number of degree-n permutations of order dividing 7.

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A061136 Number of degree-n odd permutations of order dividing 4.

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A053499 Number of degree-n permutations of order dividing 9.

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A053502 Number of degree-n permutations of order dividing 12.

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A061131 Number of degree-n even permutations of order dividing 8.

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A061140 Number of degree-n odd permutations of order exactly 8.