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 1-7 of 7 results.

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).

A181951 Number of cyclic subgroups of prime order in the Alternating Group A_n.

Original entry on oeis.org

0, 0, 1, 7, 31, 121, 526, 2227, 9283, 54931, 694156, 6104011, 76333687, 872550043, 7491293356, 49469173951, 1571562887071, 24729107440927, 584036983443568, 8662243014551731, 87570785839885951, 1147293350653737211, 66175018194591458692, 1378758190497550145383
Offset: 1

Views

Author

Olivier Gérard, Apr 03 2012

Keywords

Links

Crossrefs

Cf. A001465, A181955, A186202 (symmetric group).

Programs

  • Mathematica
    a[n_] := Sum[If[PrimeQ[p], Sum[If[p > 2 || Mod[k, 2] == 0, n!/(k!*(n - k*p)!*p^k)/(p - 1), 0], {k, 1, n/p}], 0], {p, 2, n}];
Array[a, 24] (* Jean-François Alcover, Jul 06 2018, after Andrew Howroyd *)
  • PARI
    a(n)={sum(p=2, n, if(isprime(p), sum(k=1, n\p, if(p>2||k%2==0, n!/(k!*(n-k*p)!*p^k)))/(p-1)))}

Formula

a(n) = A186202(n) - A001465(n). - Andrew Howroyd, Jul 04 2018

Extensions

Terms a(9) and beyond from Andrew Howroyd, Jul 04 2018

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

A181955 Weighted sum of all cyclic subgroups of prime order in the Alternating group.

Original entry on oeis.org

0, 0, 3, 18, 90, 390, 2205, 10878, 45318, 256350, 5530305, 55869330, 865551258, 9892489698, 78223384785, 470010394350, 24530527675230, 409760923017198, 10595007772540113, 160826214447439770, 1585844008081570650, 16787211082925012730, 1362379219330719093273
Offset: 1

Views

Author

Olivier Gérard, Apr 03 2012

Keywords

Comments

Sum of p for all p-subgroups in Alt_n.

Links

Crossrefs

Cf. A181951 (number of such subgroups).
Cf. A181954 (symmetric case).
Cf. A001465.

Programs

  • PARI
    a(n)={sum(p=2, n, if(isprime(p), sum(k=1, n\p, if(p>2||k%2==0, n!/(k!*(n-k*p)!*p^k)))*p/(p-1)))} \\ Andrew Howroyd, Jul 03 2018

Formula

a(n) = A181954(n) - 2*A001465(n). - Andrew Howroyd, Jul 03 2018

Extensions

Terms a(9) and beyond from Andrew Howroyd, Jul 03 2018

A051684 Auxiliary sequence for calculation of number of even permutations of degree n and order exactly 2.

Original entry on oeis.org

0, -1, -3, -3, 5, 15, -21, -133, 27, 1215, 935, -12441, -23673, 138047, 469455, -1601265, -9112561, 18108927, 182135007, -161934625, -3804634785, -404007681, 83297957567
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, A051685.

Formula

a(n) = c(n, 2), 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.
a(n)=2*A048099(n)-A001189(n)=A048099(n)-A001465(n) a(n)=(-1)^n*A001464(n)-1 a(n)=a(n-1)-(n-1)*(a(n-2)+1) E.g.f.: -e^x+e^(x-(1/2)*x^2) - Matthew J. White (mattjameswhite(AT)hotmail.com), Mar 02 2006
a(n) = Sum((-1)^j*n!/(2^j*j!*(n-2*j)!),j=1..floor(n/2)). - Vladeta Jovovic, Mar 06 2006

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).
Showing 1-7 of 7 results.

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