A293588 -id:A293588 - cp's OEIS frontend

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

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

A118934 E.g.f.: exp(x + x^4/4).

Original entry on oeis.org

1, 1, 1, 1, 7, 31, 91, 211, 1681, 12097, 57961, 209881, 1874071, 17842111, 117303187, 575683291, 5691897121, 65641390081, 544238393041, 3362783785777, 36455473647271, 485442581801311, 4828464958268491, 35900587138847971, 423276450114749617, 6318491163509870401

Offset: 0

Paul D. Hanna, May 06 2006

nonn

Equals row sums of triangle A118933.

These are the telephone numbers T^(4)n of [Artioli et al., p. 7]. - _Eric M. Schmidt, Oct 12 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Marcello Artioli, Giuseppe Dattoli, Silvia Licciardi, and Simonetta Pagnutti, Motzkin Numbers: an Operational Point of View, arXiv:1703.07262 [math.CO], 2017.
Vaclav Kotesovec, Graph - asymptotic (20000 terms)

Sequences with e.g.f. exp(x + x^m/m): A000079 (m=1), A000085 (m=2), A001470 (m=3), this sequence (m=4), A052501 (m=5), A293588 (m=6), A053497 (m=7).

Cf. A118933.

F:=Factorial; [(&+[F(n)/(4^j*F(j)*F(n-4*j)): j in [0..Floor(n/4)]]): n in [0..30]]; // G. C. Greubel, Mar 07 2021

With[{nn=30},CoefficientList[Series[Exp[x+x^4/4],{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Jan 26 2013 *)
Table[Sum[n!/(4^k*k!*(n-4*k)!), {k,0,n/4}], {n,0,30}]

a(n)=if(n<0,0,if(n==0,1,a(n-1) + (n-1)*(n-2)*(n-3)*a(n-4)))

f=factorial; [sum(f(n)/(4^j*f(j)*f(n-4*j)) for j in (0..n/4)) for n in (0..30)] # G. C. Greubel, Mar 07 2021

a(n) = a(n-1) + (n-1)*(n-2)*(n-3)*a(n-4) for n>=4, with a(0)=a(1)=a(2)=a(3)=1.
a(n) ~ 1/2 * n^(3*n/4) * exp(n^(1/4)-3*n/4). - Vaclav Kotesovec, Feb 25 2014
a(n) = Sum_{k=0..floor(n/4)} n!/(4^k*k!*(n-4*k)!). - G. C. Greubel, Mar 07 2021

cp's OEIS Frontend

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

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