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A288108
Number T(n,k) of Dyck paths of semilength n such that each level has exactly k peaks or no peaks; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
13
1, 0, 1, 0, 1, 1, 0, 3, 1, 1, 0, 5, 2, 1, 1, 0, 13, 5, 3, 1, 1, 0, 31, 15, 4, 4, 1, 1, 0, 71, 27, 10, 7, 5, 1, 1, 0, 181, 76, 36, 11, 11, 6, 1, 1, 0, 447, 196, 83, 22, 19, 16, 7, 1, 1, 0, 1111, 548, 225, 81, 32, 31, 22, 8, 1, 1, 0, 2799, 1388, 573, 235, 60, 56, 48, 29, 9, 1, 1
OFFSET
0,8
COMMENTS
T(n,k) is defined for all n,k >= 0. The triangle contains only the terms for k<=n. T(0,k) = 1 and T(n,k) = 0 if k > n > 0.
LINKS
EXAMPLE
. T(5,2) = 5: /\/\
. /\ /\ / \
. /\/\ /\/\ /\/\ / \/ \ / \
. /\/\/ \ /\/ \/\ / \/\/\ / \ / \ .
.
. T(5,3) = 3:
. /\/\/\
. /\ /\/\ /\/\ /\ / \
. / \/ \ / \/ \ / \ .
.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 3, 1, 1;
0, 5, 2, 1, 1;
0, 13, 5, 3, 1, 1;
0, 31, 15, 4, 4, 1, 1;
0, 71, 27, 10, 7, 5, 1, 1;
0, 181, 76, 36, 11, 11, 6, 1, 1;
MAPLE
b:= proc(n, k, j) option remember; `if`(n=j, 1, add(
b(n-j, k, i)*(binomial(j-1, i-1)+binomial(i, k)
*binomial(j-1, i-1-k)), i=1..min(j+k, n-j)))
end:
T:= (n, k)-> b(n, k$2):
seq(seq(T(n, k), k=0..n), n=0..14);
MATHEMATICA
b[n_, k_, j_] := b[n, k, j] = If[n == j, 1, Sum[b[n - j, k, i]*(Binomial[j - 1, i - 1] + Binomial[i, k]*Binomial[j - 1, i - 1 - k]), {i, 1, Min[j + k, n - j]}]];
T[n_, k_] := b[n, k, k];
Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 25 2018, translated from Maple *)
CROSSREFS
Row sums give A288109.
T(2n,n) gives A156043.
Sequence in context: A356115 A363756 A130160 * A287822 A162169 A216954
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jun 05 2017
STATUS
approved