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A329945 Number of permutations of [n] whose cycle lengths avoid squares.

%I A329945 #19 Oct 31 2021 07:46:57
%S A329945 1,0,1,2,3,44,175,1434,12313,59912,1057761,9211850,118785931,
%T A329945 1702959972,21390805423,339381890834,4027183717425,89818053205904,
%U A329945 1477419923299393,28377482210884242,608128083110593171,11954214606663753500,269933818505222203311
%N A329945 Number of permutations of [n] whose cycle lengths avoid squares.
%H A329945 Alois P. Heinz, <a href="/A329945/b329945.txt">Table of n, a(n) for n = 0..450</a>
%H A329945 David Harry Richman and Andrew O'Desky, <a href="https://arxiv.org/abs/2012.04615">Derangements and the p-adic incomplete gamma function</a>, arXiv:2012.04615 [math.NT], 2020.
%F A329945 a(n) mod 2 = 1 - (n mod 2) = A059841(n).
%F A329945 a(n) mod 10 = period 10: repeat [1,0,1,2,3,4,5,4,3,2] = A271751(n-1) for n>0.
%p A329945 a:= proc(n) option remember; `if`(n=0, 1, add(`if`(issqr(j), 0,
%p A329945       a(n-j)*binomial(n-1, j-1)*(j-1)!), j=1..n))
%p A329945     end:
%p A329945 seq(a(n), n=0..25);
%t A329945 a[n_] := a[n] = If[n == 0, 1, Sum[If[IntegerQ@Sqrt[j], 0,
%t A329945      a[n-j] Binomial[n-1, j-1] (j-1)!], {j, 1, n}]];
%t A329945 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Oct 31 2021, after _Alois P. Heinz_ *)
%Y A329945 Cf. A000290, A059841, A205801, A271751, A329944.
%K A329945 nonn
%O A329945 0,4
%A A329945 _Alois P. Heinz_, Nov 24 2019