OFFSET
0,5
COMMENTS
T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k <= n. T(n,k) = 0 for k > n.
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
EXAMPLE
T(3,1) = 3: {aaa}, {aa,a}, {a,a,a}.
T(3,2) = 5: {aab}, {aba}, {baa}, {ab,a}, {ba,a}.
T(3,3) = 6: {abc}, {acb}, {bac}, {bca}, {cab}, {cba}.
Triangle T(n,k) begins:
1;
0, 1;
0, 2, 2;
0, 3, 5, 6;
0, 5, 20, 18, 24;
0, 7, 46, 86, 84, 120;
0, 11, 137, 347, 456, 480, 720;
0, 15, 313, 1216, 2136, 2940, 3240, 5040;
0, 22, 836, 4253, 11128, 15300, 22200, 25200, 40320;
...
MAPLE
b:= proc(n, i, t) option remember; `if`(t=1, 1/n!,
add(b(n-j, j, t-1)/j!, j=i..n/t))
end:
g:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), n!*b(n, 0, k)):
A:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*
g(d, k), d=numtheory[divisors](j))*A(n-j, k), j=1..n)/n)
end:
T:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..n), n=0..12);
MATHEMATICA
b[n_, i_, t_] := b[n, i, t] = If[t == 1, 1/n!,
Sum[b[n - j, j, t - 1]/j!, {j, i, n/t}]];
g[n_, k_] := If[k == 0, If[n == 0, 1, 0], n!*b[n, 0, k]];
A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[Sum[d*
g[d, k], {d, Divisors[j]}]*A[n - j, k], {j, 1, n}]/n];
T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 09 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 20 2018
STATUS
approved