login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A288749 Number of Dyck paths of semilength n such that the maximal number of peaks per level equals eight. 2
1, 1, 19, 84, 461, 2222, 10577, 48943, 222627, 997735, 4417674, 19359659, 84099436, 362570722, 1552681071, 6609823112, 27989970166, 117967914457, 495087382572, 2069827499508, 8623283249034, 35811917284318, 148289870077879, 612382134256433, 2522591250558641 (list; graph; refs; listen; history; text; internal format)
OFFSET

8,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 8..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, 8)-g(n, 7):

seq(a(n), n=8..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, 8] - g[n, 7], {n, 8, 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, 8) - g(n, 7)

print map(a, xrange(8, 36)) # Indranil Ghosh, Aug 08 2017

CROSSREFS

Column k=8 of A287822.

Cf. A000108.

Sequence in context: A036564 A062639 A209369 * A039609 A063496 A027848

Adjacent sequences:  A288746 A288747 A288748 * A288750 A288751 A288752

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Jun 14 2017

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 19 22:28 EDT 2018. Contains 316378 sequences. (Running on oeis4.)