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A070947 Number of permutations on n letters that have only cycles of length 6 or less.

%I A070947 #30 May 09 2020 12:43:56
%S A070947 1,1,2,6,24,120,720,4320,29520,225360,1890720,17169120,166112640,
%T A070947 1680462720,18189031680,209008512000,2532028896000,32143053484800,
%U A070947 425585741760000,5865854258188800,84489178710067200,1266667808011315200,19700712491727974400
%N A070947 Number of permutations on n letters that have only cycles of length 6 or less.
%H A070947 Alois P. Heinz, <a href="/A070947/b070947.txt">Table of n, a(n) for n = 0..514</a>
%H A070947 P. L. Krapivsky, J. M. Luck, <a href="https://arxiv.org/abs/1902.04365">Coverage fluctuations in theater models</a>, arXiv:1902.04365 [cond-mat.stat-mech], 2019.
%H A070947 R. Petuchovas, <a href="https://arxiv.org/abs/1611.02934">Asymptotic analysis of the cyclic structure of permutations</a>, arXiv:1611.02934 [math.CO], p. 6, 2016.
%F A070947 E.g.f.: exp(x+1/2*x^2+1/3*x^3+1/4*x^4+1/5*x^5+1/6*x^6).
%p A070947 with(combstruct):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, m>=card))}, labeled]; end: A:=a(6):seq(count(A, size=n), n=0..21); # _Zerinvary Lajos_, Jun 11 2008
%p A070947 # second Maple program:
%p A070947 a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)
%p A070947       *binomial(n-1, j-1)*(j-1)!, j=1..min(n, 6)))
%p A070947     end:
%p A070947 seq(a(n), n=0..25);  # _Alois P. Heinz_, Dec 28 2017
%t A070947 terms = 22; CoefficientList[Exp[-Log[1-x] + O[x]^7 // Normal] + O[x]^terms, x]*Range[0, terms-1]! (* _Jean-François Alcover_, Dec 28 2017 *)
%o A070947 (Python)
%o A070947 from sympy.core.cache import cacheit
%o A070947 from sympy import binomial, factorial as f
%o A070947 @cacheit
%o A070947 def a(n): return 1 if n==0 else sum(a(n-j)*binomial(n - 1, j - 1)*f(j - 1) for j in range(1, min(n, 6)+1))
%o A070947 print([a(n) for n in range(31)]) # _Indranil Ghosh_, Dec 29 2017, after _Alois P. Heinz_
%Y A070947 Cf. A057693.
%K A070947 nonn
%O A070947 0,3
%A A070947 _N. J. A. Sloane_ and Sharon Sela, May 18 2002