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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
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
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.
Sequence in context: A182952 A108209 A111773 * A195415 A336271 A181084
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 22 2017
STATUS
approved