cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A061130 Number of degree-n even permutations of order dividing 6.

Original entry on oeis.org

1, 1, 1, 3, 12, 36, 126, 666, 6588, 44892, 237996, 2204676, 26370576, 219140208, 1720782792, 19941776856, 234038005776, 2243409386256, 23225205107088, 295070141019312, 4303459657780416, 55200265166477376, 660776587455193056
Offset: 0

Views

Author

Vladeta Jovovic, Apr 14 2001

Keywords

Links

Crossrefs

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

Formula

E.g.f.: 1/2*exp(x + 1/2*x^2 + 1/3*x^3 + 1/6*x^6) + 1/2*exp(x - 1/2*x^2 + 1/3*x^3 - 1/6*x^6).

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

Views

Author

N. J. A. Sloane, Jan 15 2000

Keywords

References

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

Links

Crossrefs

Cf. A000085, A001470, A001472, A053495-A053505, A005388.
Column k=10 of A008307.

Programs

  • Magma
    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
  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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
  • Sage
    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

Formula

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

Views

Author

N. J. A. Sloane, Jan 15 2000

Keywords

References

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

Links

Crossrefs

Cf. A000085, A001470, A001472, A053495-A053505, A005388.
Column k=11 of A008307.

Programs

  • Magma
    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
  • Maple
    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
  • Mathematica
    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 *)
  • Maxima
    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 */
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace( exp(x +x^11/11) )) \\ G. C. Greubel, May 15 2019
  • Sage
    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

Formula

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

Views

Author

N. J. A. Sloane, Jan 15 2000

Keywords

Comments

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

References

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

Links

Crossrefs

Cf. A000085, A001470, A001472, A053495-A053505, A005388.
Column k=16 of A008307.

Programs

  • Magma
    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
  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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
  • Sage
    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

Formula

E.g.f.: exp(x + x^2/2 + x^4/4 + x^8/8 + x^16/16).

A061134 Number of degree-n even permutations of order exactly 8.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 226800, 2494800, 29937600, 259459200, 1816214400, 10897286400, 301491590400, 4419628012800, 51209462304000, 482551041772800, 6979977625420800, 92611036249804800, 2078225819199129600
Offset: 1

Views

Author

Vladeta Jovovic, Apr 14 2001

Keywords

Crossrefs

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

Formula

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).

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

Views

Author

Vladeta Jovovic, Apr 14 2001

Keywords

Links

Crossrefs

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

Programs

  • Magma
    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
  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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
  • Sage
    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

Formula

E.g.f.: exp(x + x^3/3)*sinh(x^2/2 + x^6/6).
Linear recurrence of order 12 whose coefficients are polynomials in n of degree up to 15: see link. - Robert Israel, Jul 13 2018

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

Views

Author

Vladeta Jovovic, Apr 14 2001

Keywords

Crossrefs

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

Formula

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

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

Views

Author

Vladeta Jovovic, Apr 14 2001

Keywords

Crossrefs

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

Formula

E.g.f.: - 1/2*exp(x + 1/2*x^2) + 1/2*exp(x - 1/2*x^2) + 1/2*exp(x + 1/2*x^2 + 1/3*x^3 + 1/6*x^6) - 1/2*exp(x - 1/2*x^2 + 1/3*x^3 - 1/6*x^6).

A061122 Number of degree-n permutations of order exactly 8.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 5040, 45360, 453600, 3326400, 39916800, 363242880, 3874590720, 34767532800, 567177811200, 6897521030400, 98241008785920, 1138935652807680, 18952720774041600, 258251731634534400
Offset: 1

Views

Author

Vladeta Jovovic, Apr 14 2001

Keywords

Crossrefs

Cf. A000085, A001470, A001472, A052501, A053496-A053505, A001189, A001471, A001473, A061121-A061128.

Formula

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

A061123 Number of degree-n permutations of order exactly 9.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 40320, 403200, 2217600, 26611200, 259459200, 1695133440, 16345929600, 161902540800, 1208560953600, 50830132953600, 866513503215360, 8470676211379200, 166891791625977600, 2699606616475507200
Offset: 1

Views

Author

Vladeta Jovovic, Apr 14 2001

Keywords

Crossrefs

Cf. A000085, A001470, A001472, A052501, A053496-A053505, A001189, A001471, A001473, A061121-A061128.

Formula

E.g.f.: -exp(x+1/3*x^3)+exp(x+1/3*x^3+1/9*x^9).
Previous Showing 21-30 of 34 results. Next

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