A001470 -id:A001470 - cp's OEIS frontend

A008307 Table T(n,k) giving number of permutations of [1..n] with order dividing k, read by antidiagonals.

1, 1, 1, 1, 2, 1, 1, 4, 1, 1, 1, 10, 3, 2, 1, 1, 26, 9, 4, 1, 1, 1, 76, 21, 16, 1, 2, 1, 1, 232, 81, 56, 1, 6, 1, 1, 1, 764, 351, 256, 25, 18, 1, 2, 1, 1, 2620, 1233, 1072, 145, 66, 1, 4, 1, 1, 1, 9496, 5769, 6224, 505, 396, 1, 16, 3, 2, 1, 1, 35696, 31041, 33616, 1345, 2052, 1, 56, 9, 4, 1, 1

Offset: 1

N. J. A. Sloane

Solutions to x^k = 1 in Symm_n (the symmetric group of degree n).

			Array begins:
  1,   1,    1,    1,    1,     1,    1,     1, ...
  1,   2,    1,    2,    1,     2,    1,     2, ...
  1,   4,    3,    4,    1,     6,    1,     4, ...
  1,  10,    9,   16,    1,    18,    1,    16, ...
  1,  26,   21,   56,   25,    66,    1,    56, ...
  1,  76,   81,  256,  145,   396,    1,   256, ...
  1, 232,  351, 1072,  505,  2052,  721,  1072, ...
  1, 764, 1233, 6224, 1345, 12636, 5761, 11264, ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 257.
J. D. Dixon, B. Mortimer, Permutation Groups, Springer (1996), Exercise 1.2.13.

Alois P. Heinz, Antidiagonals n = 1..141, flattened
M. B. Kutler, C. R. Vinroot, On q-Analogs of Recursions for the Number of Involutions and Prime Order Elements in Symmetric Groups, JIS 13 (2010) #10.3.6, eq (5) for primes k.

Rows give A000034, A284517, A284518.
Columns give A000012, A000085, A001470, A001472, A052501, A053496, A053497, A053498, A053499, A053500, A053501, A053502, A053503, A053504, A053505.
Main diagonal gives A074759. - Alois P. Heinz, Feb 14 2013

A:= proc(n,k) option remember; `if`(n<0, 0, `if`(n=0, 1,
       add(mul(n-i, i=1..j-1)*A(n-j,k), j=numtheory[divisors](k))))
    end:
seq(seq(A(1+d-k, k), k=1..d), d=1..12); # Alois P. Heinz, Feb 14 2013
# alternative
A008307 := proc(n,m)
    local x,d ;
    add(x^d/d, d=numtheory[divisors](m)) ;
    exp(%) ;
    coeftayl(%,x=0,n) ;
    %*n! ;
end proc:
seq(seq(A008307(1+d-k,k),k=1..d),d=1..12) ; # R. J. Mathar, Apr 30 2017

t[n_ /; n >= 0, k_ /; k >= 0] := t[n, k] = Sum[(n!/(n - d + 1)!)*t[n - d, k], {d, Divisors[k]}]; t[, ] = 1; Flatten[ Table[ t[n - k, k], {n, 0, 12}, {k, 1, n}]] (* Jean-François Alcover, Dec 12 2011, after given formula *)

T(n+1,k) = Sum_{d|k} (n)_(d-1)*T(n-d+1,k), where (n)_i = n!/(n - i)! = n*(n - 1)*(n - 2)*...*(n - i + 1) is the falling factorial.

E.g.f. for n-th row: Sum_{n>=0} T(n,k)*t^n/n! = exp(Sum_{d|k} t^d/d).

More terms from Vladeta Jovovic, Apr 13 2001

A005388 Number of degree-n permutations of order a power of 2.

Original entry on oeis.org

1, 1, 2, 4, 16, 56, 256, 1072, 11264, 78976, 672256, 4653056, 49810432, 433429504, 4448608256, 39221579776, 1914926104576, 29475151020032, 501759779405824, 6238907914387456, 120652091860975616, 1751735807564578816, 29062253310781161472, 398033706586943258624

Offset: 0

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

Differs from A053503 first at n=32. - Alois P. Heinz, Feb 14 2013

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.

Alois P. Heinz, Table of n, a(n) for n = 0..200
J. M. Møller, Euler characteristics of equivariant subcategories, arXiv preprint arXiv:1502.01317, 2015. See page 20.
L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.
A. Recski, Enumerating partitional matroids, Preprint.
A. Recski & N. J. A. Sloane, Correspondence, 1975

Cf. A000085, A001470, A001472, A053495, A053496, A053497, A053498, A053499.

Cf. A053500, A053501, A053502, A053503, A053504, A053505.

R:=PowerSeriesRing(Rationals(), 40);
f:= func< x | Exp( (&+[x^(2^j)/2^j: j in [0..14]]) ) >;
Coefficients(R!(Laplace( f(x) ))); // G. C. Greubel, Nov 17 2022

a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
       add(mul(n-i, i=1..2^j-1)*a(n-2^j), j=0..ilog2(n))))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 14 2013

max = 23; CoefficientList[ Series[ Exp[ Sum[x^2^m/2^m, {m, 0, max}]], {x, 0, max}], x]*Range[0, max]! (* Jean-François Alcover, Sep 10 2013 *)

def f(x): return exp(sum(x^(2^j)/2^j for j in range(15)))
def A005388_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).egf_to_ogf().list()
A005388_list(40) # G. C. Greubel, Nov 17 2022

E.g.f.: exp(Sum_{m>=0} x^(2^m)/2^m).

E.g.f.: 1/Product_{k>=1} (1 - x^(2*k-1))^(mu(2*k-1)/(2*k-1)), where mu() is the Moebius function. - Seiichi Manyama, Jul 06 2024

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

Vladeta Jovovic, Apr 14 2001

			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

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..491
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.

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 *)

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

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

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.: 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

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.: 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

Vladeta Jovovic, Apr 14 2001

Alois P. Heinz, Table of n, a(n) for n = 0..570
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.

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

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 *)

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

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

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

A190865 E.g.f. exp(x+x^3/6).

Original entry on oeis.org

1, 1, 1, 2, 5, 11, 31, 106, 337, 1205, 5021, 20186, 86461, 417847, 1992355, 9860306, 53734241, 292816841, 1633818457, 9855157330, 59926837141, 370352343971, 2439935383271, 16283034762842, 109982177787505, 783404343570301, 5668314772422901, 41412522553362026

Offset: 0

Vladimir Kruchinin, May 22 2011

nonn

a(n) is the number of set partitions of {1,2,...,n} such that the size of each block divides 3. - Geoffrey Critzer, Sep 23 2011

			a(0) = 1 because (vacuously) all sizes of the blocks in the unique set partition of {} divide 3.
a(4) = 5 because there are 5 such set partitions of {1,2,3,4}: ({1},{2,3,4}) ({2},{1,3,4}) ({3},{1,2,4}) ({4},{1,2,3}) ({1},{2},{3},{4}).

Alois P. Heinz, Table of n, a(n) for n = 0..666

Cf. A001470.

Column k=3 of A275422.

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j>n, 0, a(n-j)*binomial(n-1, j-1)), j=[1, 3]))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 27 2016

Range[0, 25]! CoefficientList[Series[Exp[x + x^3/6] , {x, 0, 25}], x]

a(n):=n!*sum(1/((k)!*(n-3*k)!*6^(k)),k,0,n/3);

a(n) = n!*sum(k=0..n/3, 1/((k)!*(n-3*k)!*6^(k))), n>0, a(0)=1.
E.g.f.: E(0)/2, where E(k)= 1 + 1/(1 - x*(6+x^2)/(x*(6+x^2)+ 6*(k+1)/E(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013
Recurrence: 2*a(n) = 2*a(n-1) + (n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Jun 27 2013
a(n) ~ n^(2*n/3) * exp(-2*n/3+(2*n)^(1/3)) / (sqrt(3)*2^(n/3)) * (1 - 2^(2/3)/(6*n^(1/3)) + 13*2^(1/3)/(36*n^(2/3))). - Vaclav Kotesovec, Jun 27 2013
a(n) = hypergeom([-n/3, (1 - n)/3, (2 - n)/3], [], -9/2). - Peter Luschny, Jun 04 2021

A053497 Number of degree-n permutations of order dividing 7.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 721, 5761, 25921, 86401, 237601, 570241, 1235521, 892045441, 13348249201, 106757164801, 604924594561, 2722120577281, 10344007402561, 34479959558401, 24928970490633601, 546446134633639681, 6281586217487489041, 50248618811434961281

Offset: 0

N. J. A. Sloane, Jan 15 2000

nonn

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.

Alois P. Heinz, Table of n, a(n) for n = 0..200
L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.

Sequences with e.g.f. exp(x + x^m/m): A000079 (m=1), A000085 (m=2), A001470 (m=3), A118934 (m=4), A052501 (m=5), A293588 (m=6), this sequence (m=7).
Cf. A000085, A001470, A001472, A005388, A053495 - A053505, A261427.
Column k=7 of A008307.

R:=PowerSeriesRing(Rationals(), 31); Coefficients(R!(Laplace( Exp(x + x^7/7) ))); // G. C. Greubel, May 14 2019, Mar 07 2021

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=[1, 7])))
    end:
seq(a(n), n=0..25); # Alois P. Heinz, Feb 14 2013

CoefficientList[Series[Exp[x+x^7/7], {x, 0, 24}], x]*Range[0, 24]! (* Jean-François Alcover, Mar 24 2014 *)

my(x='x+O('x^30)); Vec(serlaplace( exp(x+x^7/7) )) \\ G. C. Greubel, May 14 2019

f=factorial; [sum(f(n)/(7^j*f(j)*f(n-7*j)) for j in (0..n/7)) for n in (0..30)] # G. C. Greubel, May 14 2019

E.g.f.: exp(x + x^7/7).

a(n) = Sum_{k=0..floor(n/7)} n!/(7^k*k!*(n-7*k)!). - G. C. Greubel, Mar 07 2021

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

Vladeta Jovovic, Apr 14 2001

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, A001465, A061136 - A061140.

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).

A362043 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/3)} (k/6)^j * binomial(n-2j,j)/(n-2j)!.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 5, 1, 1, 1, 1, 4, 9, 11, 1, 1, 1, 1, 5, 13, 21, 31, 1, 1, 1, 1, 6, 17, 31, 81, 106, 1, 1, 1, 1, 7, 21, 41, 151, 351, 337, 1, 1, 1, 1, 8, 25, 51, 241, 736, 1233, 1205, 1, 1, 1, 1, 9, 29, 61, 351, 1261, 2689, 5769, 5021, 1

Offset: 0

Seiichi Manyama, Apr 15 2023

			Square array begins:
  1,  1,  1,   1,   1,   1,   1, ...
  1,  1,  1,   1,   1,   1,   1, ...
  1,  1,  1,   1,   1,   1,   1, ...
  1,  2,  3,   4,   5,   6,   7, ...
  1,  5,  9,  13,  17,  21,  25, ...
  1, 11, 21,  31,  41,  51,  61, ...
  1, 31, 81, 151, 241, 351, 481, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..2 give A000012, A190865, A001470.
Main diagonal gives A362173.
T(n,2*n) gives A362300.
T(n,6*n) gives A362301.
Cf. A359762, A362302.

T(n, k) = n!*sum(j=0, n\3, (k/6)^j/(j!*(n-3*j)!));

E.g.f. of column k: exp(x + k*x^3/6).
T(n,k) = T(n-1,k) + k * binomial(n-1,2) * T(n-3,k) for n > 2.
T(n,k) = n! * Sum_{j=0..floor(n/3)} (k/6)^j / (j! * (n-3*j)!).

cp's OEIS Frontend

A008307 Table T(n,k) giving number of permutations of [1..n] with order dividing k, read by antidiagonals.

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A005388 Number of degree-n permutations of order a power of 2.

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

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

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

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A061129 Number of degree-n even permutations of order dividing 4.

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A190865 E.g.f. exp(x+x^3/6).

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A053497 Number of degree-n permutations of order dividing 7.

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A362043 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/3)} (k/6)^j * binomial(n-2j,j)/(n-2j)!.