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A289030 Number of Dyck paths having exactly n peaks in each of the levels 1,2,3 and no other peaks. 2
1, 10, 471, 27076, 1713955, 114751470, 7969151855, 567878871304, 41247976697019, 3040572724077010, 226777538499783271, 17076122335343354700, 1296037531424347164115, 99025149551454886937590, 7609414766853344476768095, 587623058661705739915402256 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The semilengths of Dyck paths counted by a(n) are elements of the integer interval [3*n+2, 6*n] for n>0.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..523

Wikipedia, Counting lattice paths

EXAMPLE

. a(1) = 10:

.

.        /\        /\          /\        /\

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

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

.

.                /\        /\                /\

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

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

.

.              /\        /\            /\

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

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

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:= n-> `if`(n=0, 1, add(b(w, n$2, 3), w=3*n+2..6*n)):

seq(a(n), n=0..15);

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_]:=If[n==0, 1, Sum[b[w, n, n, 3], {w, 3n + 2, 6n}]]; Table[a[n], {n, 0, 15}] (* Indranil Ghosh, Jul 06 2017, after maple code *)

CROSSREFS

Row n=3 of A288972.

Sequence in context: A221043 A337757 A288548 * A323205 A257133 A159533

Adjacent sequences:  A289027 A289028 A289029 * A289031 A289032 A289033

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Jun 22 2017

STATUS

approved

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Last modified May 25 21:44 EDT 2022. Contains 354071 sequences. (Running on oeis4.)