A008307 -id:A008307 - cp's OEIS frontend

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

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).

A053500 Number of degree-n permutations of order dividing 10.

Original entry on oeis.org

1, 1, 2, 4, 10, 50, 220, 1240, 6140, 32860, 602200, 5668400, 62030200, 522328600, 4487190800, 62591332000, 715163146000, 9573774122000, 105731659828000, 1187355279592000, 29205778751300000, 481597207656340000, 9086318388933400000, 132525988426667120000

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=10 of A008307.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^2/2 + x^5/5 + x^10/10) )); [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, 5, 10])))
    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, 5, 10}}]]]; 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^5/5 +x^10/10], {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^5/5 + x^10/10) )) \\ G. C. Greubel, May 15 2019

m = 30; T = taylor(exp(x + x^2/2 + x^5/5 + x^10/10), 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^5/5 + x^10/10).

A053501 Number of degree-n permutations of order dividing 11.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3628801, 43545601, 283046401, 1320883201, 4953312001, 15850598401, 44910028801, 115482931201, 274271961601, 609493248001, 1279935820801, 4644633666390681601, 106826520356358566401, 1281918194457262387201

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
Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.
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=11 of A008307.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^11/11) )); [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, 11])))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 14 2013

a[n_]:= n!*Sum[If[Mod[11*k-n, 10] == 0, Binomial[k, (11*k-n)/10]*11^((k-n)/10)/k!, 0], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Mar 20 2014, after Vladimir Kruchinin *)
With[{m = 30}, CoefficientList[Series[Exp[x +x^11/11], {x, 0, m}], x]*Range[0, m]!] (* G. C. Greubel, May 15 2019 *)

a(n):=n!*sum(if mod(11*k-n,10)=0 then binomial(k,(11*k-n)/10)*(11)^((k-n)/10)/k! else 0,k,1,n); /* Vladimir Kruchinin, Sep 10 2010 */

my(x='x+O('x^30)); Vec(serlaplace( exp(x +x^11/11) )) \\ G. C. Greubel, May 15 2019

m = 30; T = taylor(exp(x +x^11/11), 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^11/11).

a(n) = n!*Sum_{k=1..n} (if mod(11*k-n,10)=0 then C(k,(11*k-n)/10)*(11)^((k-n)/10)/k!, else 0), n>0. - Vladimir Kruchinin, Sep 10 2010

A053503 Number of degree-n permutations of order dividing 16.

Original entry on oeis.org

1, 1, 2, 4, 16, 56, 256, 1072, 11264, 78976, 672256, 4653056, 49810432, 433429504, 4448608256, 39221579776, 1914926104576, 29475151020032, 501759779405824, 6238907914387456, 120652091860975616, 1751735807564578816, 29062253310781161472, 398033706586943258624

Offset: 0

N. J. A. Sloane, Jan 15 2000

nonn

Differs from A005388 first at n=32. - Alois P. Heinz, Feb 14 2013

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=16 of A008307.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^2/2 + x^4/4 + x^8/8 + x^16/16) )); [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..2^j-1)*a(n-2^j), j=0..4)))
    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, 2^j-1}]* a[n-2^j], {j, 0, 4}]]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 19 2014, after Alois P. Heinz *)
With[{m = 30}, CoefficientList[Series[Exp[x +x^2/2 +x^4/4 +x^8/8 + x^16/16], {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^4/4 + x^8/8 + x^16/16) )) \\ G. C. Greubel, May 15 2019

m = 30; T = taylor(exp(x + x^2/2 + x^4/4 + x^8/8 + x^16/16), 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^4/4 + x^8/8 + x^16/16).

A284517 Periodic with period [1, 4, 3, 4, 1, 6] of length 6.

Original entry on oeis.org

1, 4, 3, 4, 1, 6, 1, 4, 3, 4, 1, 6, 1, 4, 3, 4, 1, 6, 1, 4, 3, 4, 1, 6, 1, 4, 3, 4, 1, 6, 1, 4, 3, 4, 1, 6, 1, 4, 3, 4, 1, 6, 1, 4, 3, 4, 1, 6, 1, 4, 3, 4, 1, 6, 1, 4, 3, 4, 1, 6, 1, 4, 3, 4, 1, 6, 1, 4, 3, 4, 1, 6, 1, 4, 3, 4, 1, 6, 1, 4, 3, 4, 1, 6, 1, 4, 3, 4, 1, 6, 1, 4, 3, 4, 1, 6, 1, 4, 3, 4, 1, 6, 1, 4, 3, 4, 1, 6

Offset: 1

R. J. Mathar, Mar 28 2017

Index entries for linear recurrences with constant coefficients, signature (-1,0,1,1).

Row 3 of A008307.

Cf. A284518.

G.f. -x*(1+5*x+7*x^2+6*x^3) / ( (x-1)*(1+x)*(1+x+x^2) ).
a(n+6) = a(n).
a(n) = (24 - ((n mod 6)^5 - 13*(n mod 6)^4 + 61*(n mod 6)^3 - 123*(n mod 6)^2 + 94*(n mod 6)))/4 for n>0. - Luce ETIENNE, Oct 19 2017

A284518 Periodic with period [1, 10, 9, 16, 1, 18, 1, 16, 9, 10, 1, 24] of length 12.

Original entry on oeis.org

1, 10, 9, 16, 1, 18, 1, 16, 9, 10, 1, 24, 1, 10, 9, 16, 1, 18, 1, 16, 9, 10, 1, 24, 1, 10, 9, 16, 1, 18, 1, 16, 9, 10, 1, 24, 1, 10, 9, 16, 1, 18, 1, 16, 9, 10, 1, 24, 1, 10, 9, 16, 1, 18, 1, 16, 9, 10, 1, 24, 1, 10, 9, 16, 1, 18, 1, 16, 9, 10, 1, 24, 1, 10, 9, 16, 1, 18, 1, 16

Offset: 1

R. J. Mathar, Mar 28 2017

Index entries for linear recurrences with constant coefficients, signature (-1,-1,0,1,1,1).

Row 4 of A008307.

Cf. A284517.

a(n+12)=a(n).

G.f.: -x*(1+11*x+20*x^2+35*x^3+25*x^4+24*x^5) / ( (x-1)*(1+x)*(1+x+x^2)*(x^2+1) ). - R. J. Mathar, Mar 28 2017

cp's OEIS Frontend

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

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

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

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A053500 Number of degree-n permutations of order dividing 10.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A053501 Number of degree-n permutations of order dividing 11.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Maxima

PARI

Sage

Formula

A053503 Number of degree-n permutations of order dividing 16.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A284517 Periodic with period [1, 4, 3, 4, 1, 6] of length 6.

Views

Author

Keywords

Links

Crossrefs

Formula