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A053500 Number of degree-n permutations of order dividing 10. 2

%I

%S 1,1,2,4,10,50,220,1240,6140,32860,602200,5668400,62030200,522328600,

%T 4487190800,62591332000,715163146000,9573774122000,105731659828000,

%U 1187355279592000,29205778751300000,481597207656340000,9086318388933400000,132525988426667120000

%N Number of degree-n permutations of order dividing 10.

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

%H Alois P. Heinz, <a href="/A053500/b053500.txt">Table of n, a(n) for n = 0..200</a>

%H L. Moser and M. Wyman, <a href="http://dx.doi.org/10.4153/CJM-1955-020-0">On solutions of x^d = 1 in symmetric groups</a>, Canad. J. Math., 7 (1955), 159-168.

%F E.g.f.: exp(x + x^2/2 + x^5/5 + x^10/10).

%p a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,

%p add(mul(n-i, i=1..j-1)*a(n-j), j=[1, 2, 5, 10])))

%p end:

%p seq(a(n), n=0..25); # _Alois P. Heinz_, Feb 14 2013

%t a[n_]:= a[n] = If[n<0, 0, If[n==0, 1, Sum[Product[n-i, {i, 1, j-1}] *a[n-j], {j, {1, 2, 5, 10}}]]]; Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Apr 24 2014, after _Alois P. Heinz_ *)

%t With[{m = 30}, CoefficientList[Series[Exp[x +x^2/2 +x^5/5 +x^10/10], {x, 0, m}], x]*Range[0, m]!] (* _G. C. Greubel_, May 15 2019 *)

%o (PARI) my(x='x+O('x^30)); Vec(serlaplace( exp(x + x^2/2 + x^5/5 + x^10/10) )) \\ _G. C. Greubel_, May 15 2019

%o (MAGMA) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^2/2 + x^5/5 + x^10/10) )); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, May 15 2019

%o (Sage) m = 30; T = taylor(exp(x + x^2/2 + x^5/5 + x^10/10), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # _G. C. Greubel_, May 15 2019

%Y Cf. A000085, A001470, A001472, A053495-A053505, A005388.

%Y Column k=10 of A008307.

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Jan 15 2000

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Last modified June 25 04:01 EDT 2021. Contains 345450 sequences. (Running on oeis4.)