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A273997 Number of endofunctions on [n] whose cycle lengths are squares.

%I A273997 #10 Jun 06 2018 03:50:06
%S A273997 1,1,3,16,131,1446,19957,329344,6315129,137942380,3382214291,
%T A273997 92014156224,2751300514987,89701699067176,3167429783609925,
%U A273997 120428877629249536,4905431165356442993,213120603686615692176,9837426739843075654819,480775495859934668704000
%N A273997 Number of endofunctions on [n] whose cycle lengths are squares.
%H A273997 Alois P. Heinz, <a href="/A273997/b273997.txt">Table of n, a(n) for n = 0..386</a>
%p A273997 b:= proc(n) option remember; local r, f, g;
%p A273997       if n=0 then 1 else r, f, g:=0, 1, 3;
%p A273997       while f<=n do r:= r+(f-1)!*b(n-f)*
%p A273997          binomial(n-1, f-1); f, g:= f+g, g+2
%p A273997       od; r fi
%p A273997     end:
%p A273997 a:= n-> add(b(j)*n^(n-j)*binomial(n-1, j-1), j=0..n):
%p A273997 seq(a(n), n=0..20);
%t A273997 b[n_] := b[n] = Module[{r, f, g}, If[n == 0, 1, {r, f, g} = {0, 1, 3}; While[f <= n, r = r + (f - 1)!*b[n - f]*Binomial[n - 1, f - 1]; {f, g} = {f + g, g + 2}]; r]];
%t A273997 a[0] = 1; a[n_] := Sum[b[j]*n^(n - j)*Binomial[n - 1, j - 1], {j, 0, n}];
%t A273997 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jun 06 2018, from Maple *)
%Y A273997 Cf. A000290, A060435, A116956, A205801, A273994, A273996, A273998.
%K A273997 nonn
%O A273997 0,3
%A A273997 _Alois P. Heinz_, Jun 06 2016