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A261317 Number of permutations sigma of [n] without fixed points such that sigma^6 = Id.

%I A261317 #16 Jul 04 2023 08:56:28
%S A261317 1,0,1,2,3,20,175,210,4585,24920,101745,1266650,13562395,48588540,
%T A261317 1082015935,9135376250,63098660625,1069777108400,13628391601825,
%U A261317 88520971388850,2134604966569075,23945393042070500,236084869688242575,4893567386193135650,72576130763294383225
%N A261317 Number of permutations sigma of [n] without fixed points such that sigma^6 = Id.
%H A261317 Alois P. Heinz, <a href="/A261317/b261317.txt">Table of n, a(n) for n = 0..500</a>
%F A261317 E.g.f.: exp(x^2*(x^4+2*x+3)/6).
%F A261317 D-finite with recurrence a(n) +(-n+1)*a(n-2) -(n-1)*(n-2)*a(n-3) -(n-5)*(n-1)*(n-2)*(n-3)*(n-4)*a(n-6)=0. - _R. J. Mathar_, Jul 04 2023
%e A261317 a(4) = 3: 2143, 3412, 4321.
%e A261317 a(5) = 20: 21453, 21534, 23154, 24513, 25431, 31254, 34152, 34521, 35124, 35412, 41523, 43251, 43512, 45132, 45213, 51432, 53214, 53421, 54123, 54231.
%p A261317 a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
%p A261317        add(mul(n-i, i=1..j-1)*a(n-j), j=[2, 3, 6])))
%p A261317     end:
%p A261317 seq(a(n), n=0..30);
%t A261317 a[n_] := a[n] = If[n < 0, 0, If[n == 0, 1, Sum[Product[n - i, {i, 1, j - 1}]*a[n - j], {j, {2, 3, 6}}]]];
%t A261317 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jun 10 2018, from Maple *)
%Y A261317 Column k=6 of A261430.
%Y A261317 Cf. A001147, A052502, A052503, A052504, A053496, A261381.
%K A261317 nonn
%O A261317 0,4
%A A261317 _Alois P. Heinz_, Aug 14 2015