OFFSET
0,4
COMMENTS
T(n,k) is defined for all n,k >= 0. The triangle contains only the terms for k <= n. T(0,k) = 1, T(n,k) = 0 for k > n > 0.
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Counting lattice paths
FORMULA
T(n,k) = Sum_{i=k..n} A288387(n,i) if k <= n.
EXAMPLE
T(4,1) = 6:
/\ /\ /\/\ /\ /\/\
/\/\/\/\ /\/\/ \ /\/ \/\ /\/ \ / \/\/\ / \/\ .
Triangle T(n,k) begins:
1;
1, 1;
2, 1, 1;
5, 3, 1, 1;
14, 6, 1, 1, 1;
42, 17, 4, 1, 1, 1;
132, 49, 14, 1, 1, 1, 1;
429, 147, 35, 5, 1, 1, 1, 1;
1430, 459, 91, 30, 1, 1, 1, 1, 1;
4862, 1476, 268, 96, 6, 1, 1, 1, 1, 1;
MAPLE
b:= proc(n, k, j) option remember; `if`(j=n, 1,
add(add(binomial(i, m)*binomial(j-1, i-1-m),
m=max(k, i-j)..i-1)*b(n-j, k, i), i=1..n-j))
end:
T:= proc(n, k) option remember; `if`(n=0, 1,
add(b(n, k, j), j=k..n))
end:
seq(seq(T(n, k), k=0..n), n=0..14);
MATHEMATICA
b[n_, k_, j_]:=b[n, k, j]=If[j==n, 1, Sum[Sum[Binomial[i, m] Binomial[j - 1, i - 1 - m], {m, Max[k, i - j], i - 1}] b[n - j, k, i], {i, n - j}]]; T[n_, k_]:=T[n, k]=If[n==0, 1, Sum[b[n, k, j], {j, k, n}]]; Table[T[n, k], {n, 0, 15}, {k, 0, n}] // Flatten (* Indranil Ghosh, Aug 09 2017 *)
PROG
(Python)
from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def b(n, k, j): return 1 if j==n else sum(sum(binomial(i, m)*binomial(j - 1, i - 1 - m) for m in range(max(k, i - j), i))*b(n - j, k, i) for i in range(1, n - j + 1))
@cacheit
def T(n, k): return 1 if n==0 else sum(b(n, k, j) for j in range(k, n + 1))
for n in range(16): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Aug 09 2017
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jun 08 2017
STATUS
approved