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A287901
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Number of Dyck paths of semilength n such that each positive level up to the highest nonempty level has at least one peak.
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4
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1, 1, 1, 3, 6, 17, 49, 147, 459, 1476, 4856, 16282, 55466, 191474, 668510, 2356944, 8380944, 30025814, 108289093, 392871484, 1432934360, 5251507624, 19329771911, 71430479820, 264914270527, 985737417231, 3679051573264, 13769781928768, 51670641652576
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graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,4
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..300
Wikipedia, Counting lattice paths
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EXAMPLE
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. a(3) = 3:
. /\ /\
. /\/\/\ /\/ \ / \/\ .
.
. a(4) = 6:
. /\ /\ /\/\ /\ /\/\
. /\/\/\/\ /\/\/ \ /\/ \/\ /\/ \ / \/\/\ / \/\ .
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MATHEMATICA
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b[n_, k_, j_]:=b[n, k, j]=If[j==n, 1, Sum[Sum[Binomial[i, m] Binomial[j - 1, i - 1 - m], {m, Max[k, i - j], i - 1}] b[n - j, k, i], {i, n - j}]]; a[n_]:=If[n==0, 1, Sum[b[n, 1, j], {j, n}]]; Table[a[n], {n, 0, 30}] (* Indranil Ghosh, Aug 09 2017 *)
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PROG
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(Python)
from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def b(n, k, j): return 1 if j==n else sum([sum([binomial(i, m)*binomial(j - 1, i - 1 - m) for m in range(max(k, i - j), i)])*b(n - j, k, i) for i in range(1, n - j + 1)])
def a(n): return 1 if n==0 else sum([b(n, 1, j) for j in range(1, n + 1)])
print([a(n) for n in range(31)]) # Indranil Ghosh, Aug 09 2017
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CROSSREFS
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Column k=1 of A288386.
Cf. A000108, A281874, A287846.
Sequence in context: A204517 A307685 A360273 * A354878 A143093 A117712
Adjacent sequences: A287898 A287899 A287900 * A287902 A287903 A287904
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KEYWORD
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nonn
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AUTHOR
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Alois P. Heinz, Jun 02 2017
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STATUS
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approved
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