A245959 -id:A245959 - cp's OEIS frontend

A246070 Number A(n,k) of endofunctions f on [2n] satisfying f^k(i) = i for all i in [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 4, 1, 2, 256, 1, 3, 16, 46656, 1, 2, 50, 216, 16777216, 1, 3, 36, 1626, 4096, 10000000000, 1, 2, 56, 1440, 83736, 100000, 8916100448256, 1, 3, 16, 2688, 84624, 6026120, 2985984, 11112006825558016, 1, 2, 70, 720, 215760, 7675200, 571350096, 105413504, 18446744073709551616

Offset: 0

Alois P. Heinz, Aug 12 2014

			Square array A(n,k) begins:
0 :            1,      1,       1,       1,        1,        1, ...
1 :            4,      2,       3,       2,        3,        2, ...
2 :          256,     16,      50,      36,       56,       16, ...
3 :        46656,    216,    1626,    1440,     2688,      720, ...
4 :     16777216,   4096,   83736,   84624,   215760,    94816, ...
5 :  10000000000, 100000, 6026120, 7675200, 24899120, 11218000, ...

Alois P. Heinz, Antidiagonals n = 0..70, flattened

Columns k=0-3 give: A085534, A062971, A245141, A245959.
Main diagonal gives A246071.
Cf. A246072 (the same for permutations).

with(numtheory): with(combinat): M:=multinomial:
b:= proc(n, k, p) local l, g; l, g:= sort([divisors(p)[]]),
      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:
A:= (n, k)-> `if`(k=0, (2*n)^(2*n), b(2*n, n, k)):
seq(seq(A(n, d-n), n=0..d), d=0..10);

multinomial[n_, k_List] := n!/Times @@ (k!); M = multinomial;
b[n_, k0_, p_] := Module[{l, g}, l = Divisors[p];
g[k_, m_, i_, t_] := g[k, m, i, t] = Module[{d, j}, d = l[[i]];
    If[i == 1, If[m == 0, 1, n^m], Sum[M[k, Join[{k - (d - t)*j},
    Table[d - t, {j}]]]/j!*If[j == 0, 1, (d - 1)!^j]*M[m, Join[{m - t*j},
    Array[t&, j]]]*g[k - (d - t)*j, m - t*j, Sequence @@
    If[d - t == 1, {i - 1, 0}, {i, t + 1}]], {j, 0, Min[k/(d - t),
    If[t == 0, {}, m/t]]}]]];
g[k0, n - k0, Length[l], 0]];
A[n_, k_] := If[k == 0, If[n == 0, 1, (2n)^(2n)], b[2*n, n, k]];
Table[A[n, d - n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 27 2016, after Alois P. Heinz, updated Jan 01 2021 *)

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.

Original entry on oeis.org

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

Alois P. Heinz, Aug 08 2014

			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;
     ...

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

Column k=0 gives A000312.
T(2n,n) gives A245959.
Main diagonal gives A001470.
Cf. A241015.

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);

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 *)

cp's OEIS Frontend

A246070 Number A(n,k) of endofunctions f on [2n] satisfying f^k(i) = i for all i in [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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