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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.

%I A245958 #14 Dec 16 2021 16:51:11
%S A245958 1,1,1,4,2,1,27,11,5,3,256,88,36,18,9,3125,925,335,141,57,21,46656,
%T A245958 12096,3912,1440,516,186,81,823543,189679,55377,18279,6003,2079,837,
%U A245958 351,16777216,3473408,924160,277824,84624,27672,10116,3690,1233
%N 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.
%H A245958 Alois P. Heinz, <a href="/A245958/b245958.txt">Rows n = 0..140, flattened</a>
%e A245958 Triangle T(n,k) begins:
%e A245958 0 :       1;
%e A245958 1 :       1,      1;
%e A245958 2 :       4,      2,     1;
%e A245958 3 :      27,     11,     5,     3;
%e A245958 4 :     256,     88,    36,    18,    9;
%e A245958 5 :    3125,    925,   335,   141,   57,   21;
%e A245958 6 :   46656,  12096,  3912,  1440,  516,  186,  81;
%e A245958 7 :  823543, 189679, 55377, 18279, 6003, 2079, 837, 351;
%e A245958      ...
%p A245958 with(combinat): M:=multinomial:
%p A245958 T:= proc(n, k) local l, g; l, g:= [1, 3],
%p A245958       proc(k, m, i, t) option remember; local d, j; d:= l[i];
%p A245958         `if`(i=1, n^m, add(M(k, k-(d-t)*j, (d-t)$j)/j!*
%p A245958          (d-1)!^j *M(m, m-t*j, t$j) *g(k-(d-t)*j, m-t*j,
%p A245958         `if`(d-t=1, [i-1, 0], [i, t+1])[]), j=0..min(k/(d-t),
%p A245958         `if`(t=0, [][], m/t))))
%p A245958       end; g(k, n-k, nops(l), 0)
%p A245958     end:
%p A245958 seq(seq(T(n, k), k=0..n), n=0..12);
%t A245958 M[n_, m_, k_List] := n!/Times @@ (Join[{m}, k]!);
%t A245958 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]];
%t A245958 Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Oct 03 2019, after _Alois P. Heinz_ *)
%Y A245958 Column k=0 gives A000312.
%Y A245958 T(2n,n) gives A245959.
%Y A245958 Main diagonal gives A001470.
%Y A245958 Cf. A241015.
%K A245958 nonn,tabl
%O A245958 0,4
%A A245958 _Alois P. Heinz_, Aug 08 2014