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A288745 Number of Dyck paths of semilength n such that the maximal number of peaks per level equals four. 2
1, 1, 11, 38, 163, 648, 2571, 10173, 40025, 156087, 605057, 2335566, 8980883, 34412583, 131431024, 500437733, 1900135511, 7196366668, 27191450135, 102522926104, 385785153584, 1448985664032, 5432879981201, 20337296148823, 76015000686028, 283720418696600 (list; graph; refs; listen; history; text; internal format)
OFFSET

4,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 4..1000

Wikipedia, Counting lattice paths

MAPLE

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, 4)-g(n, 3):

seq(a(n), n=4..35);

MATHEMATICA

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, 4] - g[n, 3], {n, 4, 35}] (* Indranil Ghosh, Aug 08 2017 *)

PROG

(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 xrange(max(0, i - j), min(k, i - 1) + 1)]) for i in xrange(1, min(j + k, n - j) + 1)])

def g(n, k): return sum([b(n, k, j) for j in xrange(1, k + 1)])

def a(n): return g(n, 4) - g(n, 3)

print map(a, xrange(4, 36)) # Indranil Ghosh, Aug 08 2017

CROSSREFS

Column k=4 of A287822.

Cf. A000108.

Sequence in context: A024202 A213775 A133258 * A103738 A045801 A162261

Adjacent sequences:  A288742 A288743 A288744 * A288746 A288747 A288748

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Jun 14 2017

STATUS

approved

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Last modified March 25 10:08 EDT 2019. Contains 321469 sequences. (Running on oeis4.)