A245911 Number of pairs of endofunctions f, g on [n] satisfying f(g^n(i)) = f(i) for all i in [n].

1, 1, 12, 207, 9184, 173225, 46097856, 729481375, 454190410752, 30607186160529, 12762075858688000, 1036636706945881151, 3080713389889966460928, 145084860487902521548921, 124325137916420574135066624, 56537825009822523196823829375

Offset: 0

Author

Alois P. Heinz, Aug 06 2014

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..100

Crossrefs

Main diagonal of A245910.

Cf. A245988.

Programs

Maple

with(combinat):
T:= proc(n, j) T(n, j):= binomial(n-1, j-1)*n^(n-j) end:
b:= proc(n, i, k) option remember; `if`(n=0 or i=1, x^n,
      expand(add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
      x^(igcd(i, k)*j)*b(n-i*j, i-1, k), j=0..n/i)))
    end:
a:= n-> add((p-> add(n^i*T(n, j)* coeff(p, x, i),
        i=0..degree(p)))(b(j$2, n)), j=0..n):
seq(a(n), n=0..20);

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, k_] := b[n, i, k] = Function[{x}, If[n == 0 || i == 1, x^n, Expand[Sum[(i - 1)!^j*multinomial[n, Join[{n - i*j}, Array[i&, j]]]/j!* x^(GCD[i, k]*j)*b[n - i*j, i - 1, k][x], {j, 0, n/i}]]]];
a[n_] := If[n == 0, 1, Sum[Binomial[n - 1, j - 1]*n^(n - j)*b[j, j, n][n], {j, 0, n}]];
a /@ Range[0, 20] (* Jean-François Alcover, Oct 03 2019, after Alois P. Heinz *)