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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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).

A261317 Number of permutations sigma of [n] without fixed points such that sigma^6 = Id.

Original entry on oeis.org

1, 0, 1, 2, 3, 20, 175, 210, 4585, 24920, 101745, 1266650, 13562395, 48588540, 1082015935, 9135376250, 63098660625, 1069777108400, 13628391601825, 88520971388850, 2134604966569075, 23945393042070500, 236084869688242575, 4893567386193135650, 72576130763294383225
Offset: 0

Views

Author

Alois P. Heinz, Aug 14 2015

Keywords

Examples

			a(4) = 3: 2143, 3412, 4321.
a(5) = 20: 21453, 21534, 23154, 24513, 25431, 31254, 34152, 34521, 35124, 35412, 41523, 43251, 43512, 45132, 45213, 51432, 53214, 53421, 54123, 54231.

Links

Crossrefs

Column k=6 of A261430.
Cf. A001147, A052502, A052503, A052504, A053496, A261381.

Programs

  • 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=[2, 3, 6])))
    end:
seq(a(n), n=0..30);
  • 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, {2, 3, 6}}]]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 10 2018, from Maple *)

Formula

E.g.f.: exp(x^2*(x^4+2*x+3)/6).
D-finite with recurrence a(n) +(-n+1)*a(n-2) -(n-1)*(n-2)*a(n-3) -(n-5)*(n-1)*(n-2)*(n-3)*(n-4)*a(n-6)=0. - R. J. Mathar, Jul 04 2023

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).

A261381 Number of permutations sigma of [n] without fixed points such that sigma^10 = Id.

Original entry on oeis.org

1, 0, 1, 0, 3, 24, 15, 504, 105, 9072, 436401, 166320, 28750491, 3243240, 1307809503, 27965161224, 52309001745, 3795543015264, 2000776242465, 324424646818272, 17268536366932851, 22708075360010040, 3974396337125445231, 1436250980764880280, 548178165969608527353
Offset: 0

Views

Author

Alois P. Heinz, Aug 17 2015

Keywords

Examples

			a(4) = 3: 2143, 3412, 4321:
a(5) = 24: 23451, 23514, 24153, 24531, 25134, 25413, 31452, 31524, 34251, 34512, 35214, 35421, 41253, 41532, 43152, 43521, 45123, 45231, 51234, 51423, 53124, 53412, 54132, 54213.
a(6) = 15: 214365, 215634, 216543, 341265, 351624, 361542, 432165, 456123, 465132, 532614, 546213, 564312, 632541, 645231, 654321.

Links

Crossrefs

Column k=10 of A261430.
Cf. A001147, A052502, A052503, A052504, A053496, A261317.

Programs

  • 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=[2, 5, 10])))
    end:
seq(a(n), n=0..30);

Formula

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

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

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).
Previous Showing 11-20 of 27 results. Next

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