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A288750 Number of Dyck paths of semilength n such that the maximal number of peaks per level equals nine. 2
1, 1, 21, 98, 568, 2858, 14128, 67556, 316490, 1456952, 6612520, 29652948, 131613716, 578987886, 2527351698, 10956840549, 47212399022, 202328867061, 862840720214, 3663367687951, 15491222396862, 65268041732681, 274068630138339, 1147305286307251 (list; graph; refs; listen; history; text; internal format)
OFFSET

9,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 9..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, 9)-g(n, 8):

seq(a(n), n=9..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, 9] - g[n, 8], {n, 9, 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 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, 9) - g(n, 8)

print([a(n) for n in range(9, 36)]) # Indranil Ghosh, Aug 08 2017

CROSSREFS

Column k=9 of A287822.

Cf. A000108.

Sequence in context: A264239 A200255 A212406 * A178794 A140370 A124949

Adjacent sequences:  A288747 A288748 A288749 * A288751 A288752 A288753

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Jun 14 2017

STATUS

approved

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Last modified September 17 12:40 EDT 2021. Contains 347477 sequences. (Running on oeis4.)