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
FORMULA
T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A261718(n,k-i).
EXAMPLE
A(3,2) = 12: 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a, 1a1a1b, 1a1b1a, 1a1b1b, 1b1a1a, 1b1a1b, 1b1b1a.
Triangle T(n,k) begins:
1
0, 1;
0, 2, 3;
0, 3, 12, 10;
0, 5, 40, 81, 47;
0, 7, 104, 396, 544, 246;
0, 11, 279, 1751, 4232, 4350, 1602;
0, 15, 654, 6528, 25100, 44475, 36744, 11481;
0, 22, 1577, 23892, 136516, 369675, 512787, 352793, 95503;
...
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1, k)+`if`(i>n, 0, b(n-i, i, k)*binomial(i+k-1, k-1))))
end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, b[n - i, i, k]*Binomial[i + k - 1, k - 1]]]]; T[n_, k_] := Sum[b[n, n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 21 2017, translated from Maple *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Aug 29 2015
STATUS
approved