A245958 Number T(n,k) of endofunctions f on [n] satisfying f^3(i) = i for all i in [k]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 1, 1, 4, 2, 1, 27, 11, 5, 3, 256, 88, 36, 18, 9, 3125, 925, 335, 141, 57, 21, 46656, 12096, 3912, 1440, 516, 186, 81, 823543, 189679, 55377, 18279, 6003, 2079, 837, 351, 16777216, 3473408, 924160, 277824, 84624, 27672, 10116, 3690, 1233

Offset

0,4

Links

Alois P. Heinz, Rows n = 0..140, flattened

Example

Triangle T(n,k) begins:
0 :       1;
1 :       1,      1;
2 :       4,      2,     1;
3 :      27,     11,     5,     3;
4 :     256,     88,    36,    18,    9;
5 :    3125,    925,   335,   141,   57,   21;
6 :   46656,  12096,  3912,  1440,  516,  186,  81;
7 :  823543, 189679, 55377, 18279, 6003, 2079, 837, 351;
     ...

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:
seq(seq(T(n, k), k=0..n), n=0..12);

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]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 03 2019, after Alois P. Heinz *)

Crossrefs

Column k=0 gives A000312.

T(2n,n) gives A245959.

Main diagonal gives A001470.

Cf. A241015.

Sequence in context: A364870 A158835 A236961 * A138271 A136212 A136215

Adjacent sequences: A245955 A245956 A245957 * A245959 A245960 A245961

Keyword

nonn,tabl

Author

Alois P. Heinz, Aug 08 2014

Status

approved