A239771 Number of pairs of functions (f,g) from a size n set into itself satisfying f(x) = g(g(f(x))).

1, 1, 10, 213, 8056, 465945, 37823616, 4075467781, 560230714240, 95369455852497, 19643693349548800, 4805295720474420501, 1374890520609054683136, 454286686896040037996905, 171479277693049020232695808, 73262491601904459123264721125, 35143072854722729593790081499136

Offset: 0

Author

Chad Brewbaker, Mar 26 2014

Keywords

• nonn

Links

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

Crossrefs

Cf. A000085, A048993, A181162, A239769, A239773, A245348, A241015.

Column k=2 of A245980.

Programs

Maple

g:= proc(n) g(n):= `if`(n<2, 1, g(n-1)+(n-1)*g(n-2)) end:
a:= n-> add(binomial(n, k)*Stirling2(n, k)*k!*
        add(binomial(n-k, i)*binomial(k, i)*i!*
        g(k-i)*n^(n-k-i), i=0..min(k, n-k)), k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 18 2014

Mathematica

g[n_] := g[n] = If[n < 2, 1, g[n-1] + (n-1)*g[n-2]];
a[n_] := If[n == 0, 1, Sum[Binomial[n, k]*StirlingS2[n, k]*k!*Sum[ Binomial[n-k, i]*Binomial[k, i]*i!*g[k-i]*n^(n-k-i), {i, 0, Min[k, n-k]} ], {k, 0, n}]];
a /@ Range[0, 20] (* Jean-François Alcover, Oct 03 2019, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=0..n} C(n,k) * A048993(n,k) * k! * A245348(n,k). - Alois P. Heinz, Jul 18 2014

Extensions

a(6)-a(7) from Giovanni Resta, Mar 28 2014

a(8)-a(16) from Alois P. Heinz, Jul 18 2014