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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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

Views

Author

Vladeta Jovovic, Apr 14 2001

Keywords

References

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

Links

Crossrefs

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

Programs

  • PARI
    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

Formula

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

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).

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).

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

A051685 Auxiliary sequence for calculation of number of even permutations of degree n and order exactly 4.

Original entry on oeis.org

0, 0, 0, -6, -30, 0, 420, 2100, 6804, -20160, -376200, -2102760, -6606600, 53237184, 965306160, 5941244400, 12774059760, -305998041600, -5264368533216, -33983490935520, -16008359119200, 3139364813249280, 52132631033313600, 341037535726730304, -715693892444414400
Offset: 1

Views

Author

Vladeta Jovovic

Keywords

References

  • V. Jovovic, Some combinatorial characteristics of symmetric and alternating groups (in Russian), Belgrade, 1980, unpublished.

Crossrefs

Cf. A001189, A001473, A051684.

Formula

a(n) = c(n, 4), where c(n, d)=Sum_{k=1..n} (-1)^(k+1)*(n-1)!/(n-k)! *Sum_{l:lcm{k, l}=d} c(n-k, l), c(0, 1)=1.

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 05 2003

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 11-20 of 24 results. Next

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