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A288746
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Number of Dyck paths of semilength n such that the maximal number of peaks per level equals five.
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2
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1, 1, 13, 48, 220, 925, 3895, 16137, 66399, 271446, 1101626, 4442143, 17822176, 71191082, 283269813, 1123212251, 4439583152, 17496345670, 68765995160, 269595218881, 1054499461385, 4115767918639, 16032123369549, 62333852291879, 241935803355457, 937486479689517
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listen;
history;
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internal format)
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OFFSET
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5,3
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 5..1000
Wikipedia, Counting lattice paths
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MAPLE
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b:= proc(n, k, j) option remember; `if`(j=n, 1, add(
b(n-j, k, i)*add(binomial(i, m)*binomial(j-1, i-1-m),
m=max(0, i-j)..min(k, i-1)), i=1..min(j+k, n-j)))
end:
g:= proc(n, k) option remember; add(b(n, k, j), j=1..k) end:
a:= n-> g(n, 5)-g(n, 4):
seq(a(n), n=5..35);
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MATHEMATICA
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b[n_, k_, j_]:=b[n, k, j]=If[j==n, 1, Sum[b[n - j, k, i] Sum[Binomial[i, m] Binomial[j - 1, i - 1 - m], {m, Max[0, i - j], Min[k, i - 1]}], {i, Min[j + k, n - j]}]]; g[n_, k_]:=Sum[b[n, k, j], {j, k}]; Table[g[n, 5] - g[n, 4], {n, 5, 35}] (* Indranil Ghosh, Aug 08 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([b(n - j, k, i)*sum([binomial(i, m)*binomial(j - 1, i - 1 - m) for m in range(max(0, i - j), min(k, i - 1) + 1)]) for i in range(1, min(j + k, n - j) + 1)])
def g(n, k): return sum([b(n, k, j) for j in range(1, k + 1)])
def a(n): return g(n, 5) - g(n, 4)
print([a(n) for n in range(5, 36)]) # Indranil Ghosh, Aug 08 2017
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CROSSREFS
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Column k=5 of A287822.
Cf. A000108.
Sequence in context: A225920 A027980 A200254 * A220707 A189349 A013200
Adjacent sequences: A288743 A288744 A288745 * A288747 A288748 A288749
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KEYWORD
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nonn
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AUTHOR
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Alois P. Heinz, Jun 14 2017
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STATUS
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approved
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