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A211880 Number of permutations of n elements with no fixed points and largest cycle of length 3.

%I A211880 #19 Nov 26 2024 18:09:40
%S A211880 0,0,0,2,0,20,40,210,1120,4760,25200,157850,800800,5345340,35035000,
%T A211880 222472250,1648046400,12000388400,88529240800,720929459250,
%U A211880 5786188408000,48072795270500,424300329453000,3731123025279650,34083741984292000,323768324084205000
%N A211880 Number of permutations of n elements with no fixed points and largest cycle of length 3.
%C A211880 a(n) = A055814(n) - A123023(n). - _Vaclav Kotesovec_, Oct 09 2013
%H A211880 Alois P. Heinz, <a href="/A211880/b211880.txt">Table of n, a(n) for n = 0..250</a>
%F A211880 E.g.f.: (exp(x^3/3)-1) * exp(x^2/2).
%F A211880 Recurrence: (n-3)*a(n) = (n-1)*(2*n-5)*a(n-2) + (n-3)*(n-2)*(n-1)*a(n-3) - (n-3)*(n-2)*(n-1)*a(n-4) - (n-4)*(n-3)*(n-2)*(n-1)*a(n-5). - _Vaclav Kotesovec_, Oct 09 2013
%e A211880 a(3) = 2: (2,3,1), (3,1,2).
%p A211880 egf:= (exp(x^3/3)-1)*exp(x^2/2):
%p A211880 a:= n-> n! *coeff(series(egf, x, n+1), x, n):
%p A211880 seq(a(n), n=0..30);
%t A211880 A[n_, k_] := A[n, k] = If[n < 0, 0, If[n == 0, 1,
%t A211880      Sum[Product[n - i, {i, 1, j - 1}] A[n - j, k], {j, 2, k}]]];
%t A211880 T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]];
%t A211880 a[n_] := T[n, 3];
%t A211880 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Sep 03 2021, after _Alois P. Heinz_ in A211871 *)
%Y A211880 Column k=3 of A211871.
%Y A211880 Cf. A055814, A123023.
%K A211880 nonn
%O A211880 0,4
%A A211880 _Alois P. Heinz_, Feb 13 2013