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A070946 Number of permutations on n letters that have only cycles of length 5 or less.

%I A070946 #26 Apr 20 2025 20:12:50
%S A070946 1,1,2,6,24,120,600,3480,22800,164880,1285920,10516320,92931840,
%T A070946 877374720,8762014080,91819440000,1005716908800,11584953158400,
%U A070946 139521689740800,1748830512960000,22750446292531200,306931140411955200,4296645083802470400,62213458150660147200
%N A070946 Number of permutations on n letters that have only cycles of length 5 or less.
%H A070946 Alois P. Heinz, <a href="/A070946/b070946.txt">Table of n, a(n) for n = 0..532</a>
%H A070946 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 A070946 Vladimir Kruchinin, <a href="http://arxiv.org/abs/1009.2565">Composition of ordinary generating functions</a>, arXiv:1009.2565 [math.CO], 2010.
%H A070946 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 A070946 E.g.f.: exp(x+1/2*x^2+1/3*x^3+1/4*x^4+1/5*x^5).
%F A070946 a(n) = n!*Sum_{k=1..n} (1/k!)*Sum_{r=0..k} binomial(k,r)*Sum_{m=0..r} 2^(m-r)*binomial(r,m)*Sum_{j=0..m} binomial(m,j)*binomial(j,n-m-k-j-r)*3^(j-m)*4^(n-r-m-k-2*j)*5^(m+k+j+r-n) for n > 0. - _Vladimir Kruchinin_, Jan 26 2011
%p A070946 with(combstruct):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, m>=card))}, labeled]; end: A:=a(5):seq(count(A, size=n), n=0..21); # _Zerinvary Lajos_, Jun 11 2008
%p A070946 # second Maple program:
%p A070946 a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)
%p A070946       *binomial(n-1, j-1)*(j-1)!, j=1..min(n, 5)))
%p A070946     end:
%p A070946 seq(a(n), n=0..25);  # _Alois P. Heinz_, Jan 25 2018
%t A070946 With[{nn=30},CoefficientList[Series[Exp[x+x^2/2+x^3/3+x^4/4+ x^5/5], {x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Mar 24 2016 *)
%Y A070946 Cf. A057693.
%K A070946 nonn
%O A070946 0,3
%A A070946 _N. J. A. Sloane_ and Sharon Sela, May 18 2002