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.

Showing 1-10 of 16 results. Next

A061132 Number of degree-n even permutations of order dividing 10.

Original entry on oeis.org

1, 1, 1, 1, 4, 40, 190, 610, 1660, 13420, 174700, 1326700, 30818800, 342140800, 2534931400, 16519411000, 143752426000, 4842417082000, 73620307162000, 687934401562000, 17165461784680000, 308493094924720000, 4585953613991980000, 53843602355379220000
Offset: 0

Views

Author

Vladeta Jovovic, Apr 14 2001

Keywords

Examples

			For n=4 the a(4)=4 solutions are (1), (1, 2)(3, 4), (1, 3)(2, 4), (1, 4)(2, 3) (permutations in cyclic notation). - _Luis Manuel Rivera Martínez_, Jun 18 2019

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

  • Mathematica
    With[{nn = 22}, CoefficientList[Series[1/2 Exp[x + x^2/2 + x^5/5 + x^10/10] + 1/2 Exp[x - x^2/2 + x^5/5 - x^10/10], {x, 0, nn}], x]* Range[0, nn]!] (* Luis Manuel Rivera Martínez, Jun 18 2019 *)
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(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))) \\ Michel Marcus, Jun 18 2019

Formula

E.g.f.: 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).

A061133 Number of degree-n even permutations of order exactly 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 210, 5040, 37800, 201600, 2044350, 25530120, 213993780, 1692490800, 19767998250, 232823791200, 2235629476080, 23171222430720, 294649445112750, 4300403589581400, 55176842335916700, 660577269463243440
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.: exp(x) - 1/2*exp(x + 1/2*x^2) - 1/2*exp(x - 1/2*x^2) - exp(x + 1/3*x^3) + 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).

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

Original entry on oeis.org

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

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.

Programs

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

Formula

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

A001465 Number of degree-n odd permutations of order 2.

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 30, 126, 448, 1296, 4140, 17380, 76296, 296088, 1126216, 4940040, 23904000, 110455936, 489602448, 2313783216, 11960299360, 61878663840, 309644323296, 1587272962528, 8699800221696, 48793502304000, 268603261201600, 1487663739072576
Offset: 0

Views

Author

N. J. A. Sloane and J. H. Conway

Keywords

Comments

Number of even partitions of an n-element set avoiding the pattern 123 (see Goyt paper). - Ralf Stephan, May 08 2007

Examples

			For n=3, a(3)=3 and (1,2), (1, 3), (2, 3) are all the degree-2 odd permutations of order 2. - _Luis Manuel Rivera Martínez_, May 22 2018

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000085, A000704.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<4, (n-1)*n/2,
      ((2*n-3)*a(n-1)-(n-1)*a(n-2))/(n-2)+(n-1)*(n-3)*a(n-4))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, May 24 2018
  • Mathematica
    Table[Sum[Binomial[n , 4 i + 2] (4 i + 2)!/(2^(2 i + 1) (2 i + 1)!), {i, 0, Floor[(n - 2)/4]}], {n, 0, 22}] (* Luis Manuel Rivera Martínez, May 22 2018 *)

Formula

a(n) = Sum_{i=0..floor((n-2)/4)} C(n,4i+2)*(4i+2)!/(4i+2)!!. - Ralf Stephan, May 08 2007
Conjecture: a(n) -3*a(n-1) +3*a(n-2) -a(n-3) -(n-1)*(n-3)*a(n-4) +(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, May 30 2014
From Jianing Song, Oct 24 2020: (Start)
E.g.f.: exp(x)*sinh(x^2/2).
a(n) = A000085(n) - A000704(n). (End)

Extensions

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 11 2004

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

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.

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).

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).

A374262 Number of permutations of [n] such that the number of cycles of length k is a multiple of k for every k.

Original entry on oeis.org

1, 1, 1, 1, 4, 16, 46, 106, 316, 3564, 27756, 141516, 556656, 6678816, 73015944, 521124696, 6144018336, 75200767776, 677927254176, 4642387894944, 75217104395136, 1167068528384256, 12348761954020416, 97377968145352896, 882819252604721664, 66882151986021043200
Offset: 0

Views

Author

Alois P. Heinz, Jul 01 2024

Keywords

Examples

			a(4) = 4: (1)(2)(3)(4), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3).

Links

Crossrefs

Cf. A000704, A130268, A372545, A372579, A374292, A374320.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      add(combinat[multinomial](n, i$i*j, n-i^2*j)*
      b(n-i^2*j, i-1)*(i-1)!^(i*j)/(i*j)!, j=0..n/i^2))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);

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).
Showing 1-10 of 16 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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