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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
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
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
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 22 2017
STATUS
approved