A245959 Number of endofunctions f on [2n] satisfying f^3(i) = i for all i in [n].

1, 2, 36, 1440, 84624, 7675200, 962250624, 151851992544, 30421572307200, 7430515709340672, 2142144445293849600, 727442024443449689088, 285148327160858698469376, 127152465871110917459189760, 64226006581334387301393186816, 36322574197169989225245335040000

Offset: 0

Author

Alois P. Heinz, Aug 08 2014

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..150

Crossrefs

Cf. A245958.

Column k=3 of A246070.

Programs

Maple

with(combinat): M:=multinomial:
T:= proc(n, k) local l, g; l, g:= [1, 3],
      proc(k, m, i, t) option remember; local d, j; d:= l[i];
        `if`(i=1, n^m, add(M(k, k-(d-t)*j, (d-t)$j)/j!*
         (d-1)!^j *M(m, m-t*j, t$j) *g(k-(d-t)*j, m-t*j,
        `if`(d-t=1, [i-1, 0], [i, t+1])[]), j=0..min(k/(d-t),
        `if`(t=0, [][], m/t))))
      end; g(k, n-k, nops(l), 0)
    end:
a:= n-> T(2*n, n):
seq(a(n), n=0..20);

Mathematica

M[n_, m_, k_List] := n!/Times @@ (Join[{m}, k]!);
T[0, 0] = 1; T[n_, k_] := T[n, k] = Module[{l = {1, 3}, g}, g[k0_, m_, {i_, t_}] := g[k0, m, i, t]; g[k0_, m_, i_, t_] := g[k0, m, i, t] = Module[ {d}, d = l[[i]]; If[i == 1, n^m, Sum[M[k0, k0 - (d - t)*j, Table[(d - t), {j}]]/j!*(d - 1)!^j*M[m, m - t*j, Table[t, {j}]]*g[k0 - (d - t)*j, m - t*j, If[d - t == 1, {i - 1, 0}, {i, t + 1}]], {j, 0, Min[k0/(d - t), If[t == 0, Infinity, m/t]]}]]]; g[k, n - k, Length[l], 0]];
a[n_] := T[2 n, n];
a /@ Range[0, 20] (* Jean-François Alcover, Oct 03 2019, after Alois P. Heinz *)

Formula

a(n) = A245958(2n,n).