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A289020 Number of Dyck paths having exactly one peak in each of the levels 1,...,n and no other peaks. 4
1, 1, 2, 10, 92, 1348, 28808, 845800, 32664944, 1605553552, 97868465696, 7245440815264, 640359291096512, 66598657958731840, 8051483595083729024, 1119653568781387712128, 177465810459239319017216, 31804047327185301634148608, 6398867435594240638421950976 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The semilengths of Dyck paths counted by a(n) are elements of the integer interval [2*n-1, n*(n+1)/2] = [A060747(n), A000217(n)] for n>0.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..100

Wikipedia, Counting lattice paths

EXAMPLE

. a(2) = 2:      /\    /\

.             /\/  \  /  \/\  .

MAPLE

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

      `if`(v=1, 1, 0), `if`(v<2, 0, add(b(n-j, i, v-1)*

       i*binomial(j-1, i-2), i=1..min(j+1, n-j))))

    end:

a:= n-> `if`(n=0, 1, add(b(w, 1, n), w=2*n-1..n*(n+1)/2)):

seq(a(n), n=0..18);

MATHEMATICA

b[n_, j_, v_]:=b[n, j, v]=If[n==j, If[v==1, 1, 0], If[v<2, 0, Sum[b[n - j, i, v - 1]*i*Binomial[j - 1, i - 2], {i, Min[j + 1, n - j]}]]]; a[n_]:=If[n==0, 1, Sum[b[w, 1, n], {w, 2*n - 1, n*(n + 1)/2}]]; Table[a[n], {n, 0, 18}] (* Indranil Ghosh, Jul 06 2017, after Maple code *)

CROSSREFS

Column k=1 of A288972.

Cf. A000217, A060747, A281874, A287846.

Sequence in context: A182952 A108209 A111773 * A195415 A336271 A181084

Adjacent sequences:  A289017 A289018 A289019 * A289021 A289022 A289023

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Jun 22 2017

STATUS

approved

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Last modified August 7 04:39 EDT 2020. Contains 336274 sequences. (Running on oeis4.)