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A287987 Number of Dyck paths of semilength n such that all positive levels have the same number of peaks. 6
1, 1, 1, 3, 1, 8, 13, 13, 54, 132, 280, 547, 1219, 3904, 11107, 25082, 53777, 137751, 419831, 1257599, 3453557, 8911341, 22636845, 59890162, 172264224, 529706648, 1630328686, 4765347773, 13125989799, 35253234315, 97531470556, 287880507391, 894915519516 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..500

Wikipedia, Counting lattice paths

EXAMPLE

. a(3) = 3:                         /\        /\

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

.

. a(5) = 8:

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

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

.

.            /\        /\          /\        /\

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

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

MAPLE

b:= proc(n, k, j) option remember; `if`(n=j, 1,

       add(binomial(i, k)*binomial(j-1, i-1-k)

         *b(n-j, k, i), i=1+k..min(j+k, n-j)))

    end:

a:= n-> 1+add(b(n, j$2), j=1..n/2):

seq(a(n), n=0..33);

MATHEMATICA

b[n_, k_, j_] := b[n, k, j] = If[n == j, 1, Sum[Binomial[i, k]*Binomial[j - 1, i - 1 - k]*b[n - j, k, i], {i, 1 + k, Min[j + k, n - j]}]];

a[n_] := 1 + Sum[b[n, j, j], {j, 1, n/2}];

Table[a[n], {n, 0, 33}] (* Jean-Fran├žois Alcover, May 24 2018, translated from Maple *)

CROSSREFS

Row sums of A288318.

Cf. A000108, A287845, A287846, A287993, A288109.

Sequence in context: A019146 A102537 A131202 * A067955 A182509 A049965

Adjacent sequences:  A287984 A287985 A287986 * A287988 A287989 A287990

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Jun 03 2017

STATUS

approved

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Last modified January 19 15:37 EST 2020. Contains 331049 sequences. (Running on oeis4.)