OFFSET
0,9
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
EXAMPLE
T(3,1) = 1: 3a.
T(3,2) = 2: 2a1b, 1a1b1a.
T(3,3) = 1: 1a1b1c.
T(5,3) = 12: 3a1b1c, 2a2b1c, 2a1b1a1c, 2a1b1c1a, 2a1b1c1b, 1a1b1a1b1c, 1a1b1a1c1a, 1a1b1a1c1b, 1a1b1c1a1b, 1a1b1c1a1c, 1a1b1c1b1a, 1a1b1c1b1c.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 1, 2, 1;
0, 1, 4, 4, 1;
0, 1, 6, 12, 7, 1;
0, 1, 10, 31, 33, 11, 1;
0, 1, 14, 73, 130, 77, 16, 1;
0, 1, 21, 165, 464, 438, 157, 22, 1;
0, 1, 29, 357, 1558, 2216, 1223, 289, 29, 1;
0, 1, 41, 760, 5039, 10423, 8331, 2957, 492, 37, 1;
...
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0 or i=1, k^(n-1),
b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k)))
end:
A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, k*b(n$2, k-1))):
T:= (n, k)-> add(A(n, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..12);
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i == 1, k^(n-1), b[n, i-1, k] + If[i>n, 0, k*b[n-i, i, k]]]; A[n_, k_] := If[n == 0, 1, If[k<2, k, k*b[n, n, k-1]]]; T[n_, k_] := Sum[A[n, k-i]*(-1)^i/(i!*(k-i)!), {i, 0, k}]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 24 2017, translated from Maple *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 24 2015
STATUS
approved