A245912 Number of pairs of endofunctions f, g on [n] satisfying f(g^3(i)) = f(i) for all i in [n].

1, 1, 10, 207, 6856, 302345, 17812656, 1384059775, 131612023936, 14986421437329, 2051598980742400, 327546779949753551, 59790068922261980160, 12505503377433451819993, 2956768061598853524176896, 778675046844529953944661375, 228393818322135051214683406336

Offset

0,3

Links

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

Maple

with(combinat):
T:= proc(n, j) option remember; binomial(n-1, j-1)*n^(n-j) end:
b:= proc(n, i) 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, 3)*j)*b(n-i*j, i-1), 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)), j=0..n):
seq(a(n), n=0..20);

Mathematica

T[n_, j_] := T[n, j] = Binomial[n - 1, j - 1]*n^(n - j);
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x^n, Expand[Sum[(i - 1)!^j*Multinomial @@ Join[{n - i*j}, Table[i, {j}]]/j!* x^(GCD[i, 3]*j)*b[n - i*j, i - 1], {j, 0, n/i}]]];
a[n_] := If[n == 0, 1, Sum[With[{p = b[j, j]}, Sum[n^i*T[n, j]* Coefficient[p, x, i], {i, 0, Exponent[p, x]}]], {j, 0, n}]];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 19 2026, after Alois P. Heinz *)

Crossrefs

Column k=3 of A245910.

Sequence in context: A356973 A356960 A364989 * A245918 A385438 A368441

Adjacent sequences: A245909 A245910 A245911 * A245913 A245914 A245915

Keyword

nonn

Author

Alois P. Heinz, Aug 06 2014

Status

approved