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%I #27 Sep 08 2022 08:45:01
%S 1,0,1,2,3,20,55,210,1225,4760,26145,157850,811195,5345340,35170135,
%T 222472250,1650073425,12000388400,88563700225,720929459250,
%U 5786843137075,48072795270500,424314078763575,3731123025279650,34084058218435225,323768324084205000
%N Expansion of e.g.f.: exp(x^3/3 + x^2/2).
%C a(n) is the number of n-permutations in which all cycles have length two or three. - _Geoffrey Critzer_, Feb 21 2010
%D Miklos Bona, A Walk Through Combinatorics, World Scientific Publishing Co., 2002, page 169. - _Geoffrey Critzer_, Feb 21 2010
%H Alois P. Heinz, <a href="/A055814/b055814.txt">Table of n, a(n) for n = 0..625</a>
%F a(n) = subs(x=0, (d^n/dx^n)exp(x^3/3 + x^2/2)), n=0, 1, 2, ...
%F a(n) = (n-1)*a(n-2) + (n-1)*(n-2)*a(n-3). - _Joerg Arndt_, Oct 02 2009
%F a(n) ~ n^(2*n/3)*exp(1/18 - 2*n/3 - n^(1/3)/6 + n^(2/3)/2)/sqrt(3) * (1 + 49/(324*n^(1/3)) - 72451/(1049760*n^(2/3))). - _Vaclav Kotesovec_, Jun 26 2013
%e a(4) = 3 because there are 3 permutations of {1,2,3,4} that have cycle length two or three: (1,2)(3,4);(1,3)(2,4);(1,4)(2,3). - _Geoffrey Critzer_, Feb 21 2010
%p a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)
%p *binomial(n-1, j-1)*(j-1)!, j=2..min(3, n)))
%p end:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Jan 25 2018
%t With[{m=30}, CoefficientList[Series[Exp[x^2/2 + x^3/3], {x,0,m}], x]*Range[0, m]!] (* _Geoffrey Critzer_, Feb 21 2010 *)
%o (PARI) my(x='x+O('x^30)); Vec(serlaplace( exp(x^3/3 + x^2/2) )) \\ _G. C. Greubel_, Jan 23 2020
%o (Magma) I:=[1,0,1]; [n le 3 select I[n] else (n-2)*(Self(n-2) +(n-3)*Self(n-3)): n in [1..30]]; // _G. C. Greubel_, Jan 23 2020
%o (Sage) [factorial(n)*( exp(x^3/3 + x^2/2) ).series(x,n+1).list()[n] for n in (0..30)] # _G. C. Greubel_, Jan 23 2020
%o (GAP) a:=[1,0,1];; for n in [4..30] do a[n]:=(n-2)*(a[n-2]+(n-3)*a[n-3]); od; a; # _G. C. Greubel_, Jan 23 2020
%Y Cf. A081096.
%Y Cf. A000085, A001470. - _Joerg Arndt_, Oct 02 2009
%K nonn
%O 0,4
%A _Karol A. Penson_, Mar 05 2003
%E Improved definition, as proposed by _Joerg Arndt_, from _R. J. Mathar_, Oct 23 2009
%E a(0)=1 prepended by _Alois P. Heinz_, Jan 25 2018
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