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A288972 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. 6
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

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.

LINKS

Alois P. Heinz, Antidiagonals n = 0..26, flattened

Wikipedia, Counting lattice paths

EXAMPLE

. A(3,1) = 10:

.

.        /\        /\          /\        /\

.     /\/  \      /  \/\    /\/  \      /  \/\

.  /\/      \  /\/      \  /      \/\  /      \/\

.

.                /\        /\                /\

.           /\  /  \      /  \  /\    /\    /  \

.        /\/  \/    \  /\/    \/  \  /  \/\/    \

.

.              /\        /\            /\

.         /\  /  \      /  \    /\    /  \  /\

.        /  \/    \/\  /    \/\/  \  /    \/  \/\  .

.

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, ...

MAPLE

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);

MATHEMATICA

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 *)

CROSSREFS

Columns k=0-2 give: A000012, A289020, A289054.

Rows n=0+1,2,3 give: A000012, A176479, A289030.

Main diagonal gives A288940.

Sequence in context: A061538 A123602 A208896 * A065521 A225700 A156188

Adjacent sequences:  A288969 A288970 A288971 * A288973 A288974 A288975

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jun 20 2017

STATUS

approved

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Last modified September 21 19:57 EDT 2020. Contains 337273 sequences. (Running on oeis4.)