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

1, 1, 3, 16, 131, 1446, 19957, 329344, 6315129, 137942380, 3382214291, 92014156224, 2751300514987, 89701699067176, 3167429783609925, 120428877629249536, 4905431165356442993, 213120603686615692176, 9837426739843075654819, 480775495859934668704000

Offset: 0

Author

Alois P. Heinz, Jun 06 2016

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..386

Crossrefs

Cf. A000290, A060435, A116956, A205801, A273994, A273996, A273998.

Programs

Maple

b:= proc(n) option remember; local r, f, g;
      if n=0 then 1 else r, f, g:=0, 1, 3;
      while f<=n do r:= r+(f-1)!*b(n-f)*
         binomial(n-1, f-1); f, g:= f+g, g+2
      od; r fi
    end:
a:= n-> add(b(j)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..20);

Mathematica

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]];
a[0] = 1; a[n_] := Sum[b[j]*n^(n - j)*Binomial[n - 1, j - 1], {j, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 06 2018, from Maple *)