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A288109
Number of Dyck paths of semilength n such that all levels with peaks have exactly the same number of peaks.
3
1, 1, 2, 5, 9, 23, 56, 122, 323, 792, 2060, 5199, 13314, 35171, 94077, 249285, 662901, 1775244, 4806724, 13125887, 36107283, 99863241, 276784435, 768288783, 2143763275, 6037486060, 17171063218, 49187617277, 141512589597, 408293870713, 1181084207303
OFFSET
0,3
LINKS
MAPLE
b:= proc(n, k, j) option remember; `if`(n=j, 1, add(
b(n-j, k, i)*(binomial(j-1, i-1)+binomial(i, k)
*binomial(j-1, i-1-k)), i=1..min(j+k, n-j)))
end:
a:= n-> 1 + add(b(n, j$2), j=1..n-1):
seq(a(n), n=0..33);
MATHEMATICA
b[n_, k_, j_] := b[n, k, j] = If[n==j, 1, Sum[b[n-j, k, i]*(Binomial[j-1, i - 1] + Binomial[i, k]*Binomial[j-1, i-1-k]), {i, 1, Min[j+k, n-j]}]];
a[n_] := 1 + Sum[b[n, j, j], {j, 1, n - 1}];
Table[a[n], {n, 0, 33}] (* Jean-François Alcover, May 31 2018, from Maple *)
CROSSREFS
Row sums of A288108.
Sequence in context: A088356 A246350 A192477 * A047044 A109621 A075200
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 05 2017
STATUS
approved