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A288972
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Number A(n,k) of Dyck paths having exactly k peaks in each of the levels 1,...,n and no other peaks; square array A(n,k), n>=0, k>=0, read by antidiagonals.
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6
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1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 9, 10, 1, 1, 1, 44, 471, 92, 1, 1, 1, 225, 27076, 82899, 1348, 1, 1, 1, 1182, 1713955, 102695344, 36913581, 28808, 1, 1, 1, 6321, 114751470, 147556480375, 1565018426896, 34878248649, 845800, 1
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OFFSET
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0,9
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COMMENTS
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The semilengths of Dyck paths counted by A(n,k) are elements of the integer interval [k*n+n-1, k*n*(n+1)/2] for n,k>0.
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LINKS
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EXAMPLE
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. A(3,1) = 10:
.
. /\ /\ /\ /\
. /\/ \ / \/\ /\/ \ / \/\
. /\/ \ /\/ \ / \/\ / \/\
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. /\ /\ /\
. /\ / \ / \ /\ /\ / \
. /\/ \/ \ /\/ \/ \ / \/\/ \
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. /\ /\ /\
. /\ / \ / \ /\ / \ /\
. / \/ \/\ / \/\/ \ / \/ \/\ .
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Square array A(n,k) begins:
1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, ...
1, 2, 9, 44, 225, ...
1, 10, 471, 27076, 1713955, ...
1, 92, 82899, 102695344, 147556480375, ...
1, 1348, 36913581, 1565018426896, 81072887990665625, ...
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MAPLE
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b:= proc(n, k, j, v) option remember; `if`(n=j, `if`(v=1, 1, 0),
`if`(v<2, 0, add(b(n-j, k, i, v-1)*(binomial(i, k)*
binomial(j-1, i-1-k)), i=1..min(j+k, n-j))))
end:
A:= proc(n, k) option remember; `if`(n=0 or k=0, 1,
add(b(w, k, k, n), w=k*n+n-1..k*n*(n+1)/2))
end:
seq(seq(A(n, d-n), n=0..d), d=0..10);
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MATHEMATICA
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b[n_, k_, j_, v_]:=b[n, k, j, v]=If[n==j, If[v==1, 1, 0], If[v<2, 0, Sum[b[n - j, k, i, v - 1] Binomial[i, k] Binomial[j - 1, i - 1 - k], {i, Min[j + k, n - j]}]]]; A[n_, k_]:=A[n, k]=If[n==0 || k==0, 1, Sum[b[w, k, k, n], {w, k*n + n - 1, k*n*(n + 1)/2}]]; Table[A[n, d - n], {d, 0, 10}, {n, 0, d}] // Flatten (* Indranil Ghosh, Jul 06 2017, after Maple code *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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