A305824 Number of endofunctions on [n] whose cycle lengths are triangular numbers.

1, 1, 3, 18, 157, 1776, 24807, 413344, 8004537, 176630400, 4374300331, 120136735104, 3623854678677, 119102912981248, 4236492477409935, 162152320065532416, 6645233337842716273, 290321208589666369536, 13469914225467040015827, 661442143465113960448000

Offset: 0

Author

Alois P. Heinz, Jun 10 2018

Keywords

• nonn

Links

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

Crossrefs

Cf. A000217, A060435, A116956, A193374, A205799, A273994, A273996, A273998.

Programs

Maple

b:= proc(n) option remember; local r, f, g;
      if n=0 then 1 else r, f, g:=$0..2;
      while f<=n do r, f, g:= r+(f-1)!*
         b(n-f)*binomial(n-1, f-1), f+g, g+1
      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, 2}; While[f <= n, {r, f, g} = {r + (f - 1)!*b[n - f]*Binomial[n - 1, f - 1], f + g, g + 1}]; 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 15 2018, after Alois P. Heinz *)