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A288108
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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.
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13
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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
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OFFSET
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0,8
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COMMENTS
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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.
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LINKS
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EXAMPLE
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. 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;
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MAPLE
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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);
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MATHEMATICA
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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 *)
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CROSSREFS
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Columns k=0-10 give: A000007, A281874, A287843, A288110, A288111, A288112, A288113, A288114, A288115, A288116, A288117.
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KEYWORD
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AUTHOR
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STATUS
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approved
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