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A053497 Number of degree-n permutations of order dividing 7. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,8

REFERENCES

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

LINKS

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.

FORMULA

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

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=[1, 7])))

    end:

seq(a(n), n=0..25); # Alois P. Heinz, Feb 14 2013

MATHEMATICA

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

PROG

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

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

(Sage) 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

CROSSREFS

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.

Sequence in context: A154515 A241961 A318527 * A139154 A139165 A229049

Adjacent sequences:  A053494 A053495 A053496 * A053498 A053499 A053500

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jan 15 2000

STATUS

approved

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Last modified June 15 04:47 EDT 2021. Contains 345043 sequences. (Running on oeis4.)