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A288318
Number T(n,k) of Dyck paths of semilength n such that each positive level has exactly k peaks; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
14
1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 1, 0, 4, 3, 0, 0, 1, 0, 6, 6, 0, 0, 0, 1, 0, 8, 0, 4, 0, 0, 0, 1, 0, 24, 9, 20, 0, 0, 0, 0, 1, 0, 52, 54, 20, 5, 0, 0, 0, 0, 1, 0, 96, 138, 0, 45, 0, 0, 0, 0, 0, 1, 0, 212, 207, 16, 105, 6, 0, 0, 0, 0, 0, 1, 0, 504, 360, 200, 70, 84, 0, 0, 0, 0, 0, 0, 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
FORMULA
T(n,n) = 1.
T(n+1,n) = 0.
T(2*n+1,n) = (n+1) for n>0.
T(2*n+2,n) = A005564(n+1) for n>1.
T(3*n,n) = A000984(n) = binomial(2*n,n).
T(3*n+1,n) = 0.
T(3*n+2,n) = (n+1)^2 for n>0.
EXAMPLE
. T(5,1) = 4:
. /\ /\ /\ /\
. /\/ \ / \/\ /\/ \ / \/\
. /\/ \ /\/ \ / \/\ / \/\ .
.
. T(5,2) = 3:
. /\/\ /\/\ /\/\
. /\/\/ \ /\/ \/\ / \/\/\ .
.
Triangle T(n,k) begins:
1;
0, 1;
0, 0, 1;
0, 2, 0, 1;
0, 0, 0, 0, 1;
0, 4, 3, 0, 0, 1;
0, 6, 6, 0, 0, 0, 1;
0, 8, 0, 4, 0, 0, 0, 1;
0, 24, 9, 20, 0, 0, 0, 0, 1;
0, 52, 54, 20, 5, 0, 0, 0, 0, 1;
MAPLE
b:= proc(n, k, j) option remember;
`if`(n=j, 1, add(b(n-j, k, i)*(binomial(i, k)
*binomial(j-1, i-1-k)), i=1..min(j+k, n-j)))
end:
T:= (n, k)-> `if`(n=0, 1, 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[i, k]*Binomial[j - 1, i - 1 - k]), {i, 1, Min[j + k, n - j]}]];
T[n_, k_] := If[n == 0, 1, 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
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jun 07 2017
STATUS
approved