A001471 -id:A001471 - cp's OEIS frontend

A059593 Number of degree-n permutations of order exactly 5.

0, 0, 0, 0, 0, 24, 144, 504, 1344, 3024, 78624, 809424, 4809024, 20787624, 72696624, 1961583624, 28478346624, 238536558624, 1425925698624, 6764765838624, 189239120970624, 3500701266525624, 37764092547420624, 288099608198025624

Offset: 0

Henry Bottomley, Jan 26 2001

nonn

The number of degree-n permutations of order exactly p (where p is prime) satisfies a(n) =a(n-1) + (1+a(n-p))*(n-1)!/(n-p)! with a(n)=0 if p>n. Also a(n) = Sum_{j=1 to floor[n/p]} n!/(j!*(n-p*j)!*(p^j)).

Alois P. Heinz, Table of n, a(n) for n = 0..300

Cf. A001471, A052501.

Column k=5 of A057731. - Alois P. Heinz, Feb 16 2013

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^5/5) )); [Factorial(n-1)*b[n]-1: n in [1..m]]; // G. C. Greubel, May 14 2019

a:= proc(n) option remember;
      `if`(n<5, 0, a(n-1)+(1+a(n-5))*(n-1)!/(n-5)!)
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Jan 25 2014

Table[Sum[n!/(j!*(n-5*j)!*5^j), {j,1,Floor[n/5]}], {n,0,25}] (* G. C. Greubel, May 14 2019 *)

{a(n) = sum(j=1,floor(n/5), n!/(j!*(n-5*j)!*5^j))}; \\ G. C. Greubel, May 14 2019

m = 30; T = taylor(exp(x + x^5/5) -exp(x), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 14 2019

a(n) = a(n - 1) + (1 + a(n - 5))*(n - 1)(n - 2)(n - 3)(n - 4).
a(n) = Sum_{j=1..floor(n/5)} n!/(j!*(n - 5*j)!*(5^j)).
From G. C. Greubel, May 14 2019: (Start)
a(n) = A052501(n) - 1.
E.g.f.: exp(x + x^5/5) - exp(x). (End)

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

Vladeta Jovovic, Apr 14 2001

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

Alois P. Heinz, Table of n, a(n) for n = 0..502
Lev Glebsky, Melany Licón, Luis Manuel Rivera, On the number of even roots of permutations, arXiv:1907.00548 [math.CO], 2019.
T. Koda, M. Sato, Y. Tskegahara, 2-adic properties for the numbers of involutions in the alternating groups, J. Algebra Appl. 14 (2015), no. 4, 1550052 (21 pages).

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

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

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

Vladeta Jovovic, Apr 14 2001

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

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

Vladeta Jovovic, Apr 14 2001

Alois P. Heinz, Table of n, a(n) for n = 0..522
Lev Glebsky, Melany Licón, Luis Manuel Rivera, On the number of even roots of permutations, arXiv:1907.00548 [math.CO], 2019.

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

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).

A214003 Number of degree-n permutations of prime order.

Original entry on oeis.org

0, 1, 5, 17, 69, 299, 1805, 9099, 37331, 205559, 4853529, 49841615, 789513659, 9021065871, 70737031469, 420565124399, 22959075244095, 385032305178719, 10010973102879761, 152163983393187399, 1498273284120348539, 15639918041915598815, 1296204202723400597109

Offset: 1

Stephen A. Silver, Feb 15 2013

nonn

			The symmetric group S_5 has 25 elements of order 2, 20 elements of order 3, and 24 elements of order 5. All other elements are of nonprime order (1, 4, or 6), so a(5) = 25 + 20 + 24 = 69.

Stephen A. Silver, Table of n, a(n) for n = 1..451

Cf. A001189, A001471, A057731, A059593, A153760, A153761, A186202.

b:= proc(n,p) option remember;
      `if`(n add(b(n, ithprime(i)), i=1..numtheory[pi](n)):
seq(a(n), n=1..30);  # Alois P. Heinz, Feb 16 2013
# second Maple program:
b:= proc(n, g) option remember; `if`(n=0, `if`(isprime(g), 1, 0),
      add(b(n-j, ilcm(j, g))*(n-1)!/(n-j)!, j=1..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=1..23);  # Alois P. Heinz, Jan 19 2023

f[list_] :=Total[list]!/Apply[Times, list]/Apply[Times, Map[Length, Split[list]]!]; Table[Total[Map[f, Select[Partitions[n],PrimeQ[Apply[LCM, #]] &]]], {n, 1,23}] (* Geoffrey Critzer, Nov 08 2015 *)

a(n) = Sum_{p prime} A057731(n,p).

E.g.f.: exp(x)*Sum_{p in Primes} exp(x^p/p)-1. - Geoffrey Critzer, Nov 08 2015

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

Vladeta Jovovic, Apr 14 2001

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

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

Vladeta Jovovic, Apr 14 2001

Robert Israel, Table of n, a(n) for n = 0..522
Lev Glebsky, Melany Licón, Luis Manuel Rivera, On the number of even roots of permutations, arXiv:1907.00548 [math.CO], 2019.
Robert Israel, Recurrence for this sequence

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

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

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

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 *)

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

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

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

A336614 Number of n X n (0,1)-matrices A over the reals such that A^2 is the transpose of A.

Original entry on oeis.org

1, 2, 4, 10, 32, 112, 424, 1808, 8320, 40384, 210944, 1170688, 6783616, 41411840, 265451008, 1765520128, 12227526656, 88163295232, 656548065280, 5054719287296, 40261285543936, 330010835894272, 2783003772452864, 24166721466204160, 215318925894909952, 1966855934183800832

Offset: 0

Torlach Rush, Jul 27 2020

nonn

From Peter Luschny, Jun 04 2021: (Start)
a(n) = n! * [x^n] exp(x*(x^2 + 6)/3).
a(n) = 2*a(n - 1) + (n^2 - 3*n + 2)*a(n - 3) for n >= 3.
a(n) = Sum_{k=0..n/3} (2^(n-3*k)*n!)/(3^k*k!*(n-3*k)!).
a(n) = 2^n*hypergeom([-n/3, (1-n)/3, (2-n)/3], [], -9/8).
[The above formulas, first stated as conjectures, were proved by mjqxxxx at Mathematics Stack Exchange, see link.] (End)

			a(3) = A336174(3) + A000079(3) = 2 + 8 = 10.

mjqxxxx, Proof of conjectured formulas, Mathematics Stack Exchange.

Row sums of A344912.

Cf. A000079, A001471, A336174.

a := n -> add((2^(n - 3*k)*n!)/(3^k*k!*(n - 3*k)!), k=0..n/3):
seq(a(n), n=0..25); # Peter Luschny, Jun 05 2021

m(n, t) = matrix(n, n, i, j, (t>>(i*n+j-n-1))%2)
a(n) = sum(t = 0, 2^n^2-1, m(n, t)^2 == m(n, t)~)
for(n = 0, 9, print1(a(n), ", "))

from itertools import product
from sympy import Matrix
def A336614(n):
    c = 0
    for d in product((0,1),repeat=n*n):
        M = Matrix(d).reshape(n,n)
        if M*M == M.T:
            c += 1
    return c # Chai Wah Wu, Sep 29 2020

a(n) = A336174(n) + A000079(n).

More terms from Peter Luschny, Jun 05 2021

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

Vladeta Jovovic, Apr 14 2001

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

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

Vladeta Jovovic, Apr 14 2001

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

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).

cp's OEIS Frontend

A059593 Number of degree-n permutations of order exactly 5.

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A061131 Number of degree-n even permutations of order dividing 8.

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A061140 Number of degree-n odd permutations of order exactly 8.

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

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A214003 Number of degree-n permutations of prime order.

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

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A061137 Number of degree-n odd permutations of order dividing 6.

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A336614 Number of n X n (0,1)-matrices A over the reals such that A^2 is the transpose of A.

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