login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A287901 Number of Dyck paths of semilength n such that each positive level up to the highest nonempty level has at least one peak. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..300

Wikipedia, Counting lattice paths

EXAMPLE

. a(3) = 3:

.                   /\      /\

.      /\/\/\    /\/  \    /  \/\ .

.

. a(4) = 6:

.                    /\      /\        /\/\    /\        /\/\

.     /\/\/\/\  /\/\/  \  /\/  \/\  /\/    \  /  \/\/\  /    \/\ .

MATHEMATICA

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 *)

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([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

CROSSREFS

Column k=1 of A288386.

Cf. A000108, A281874, A287846.

Sequence in context: A099511 A204517 A307685 * A143093 A117712 A106158

Adjacent sequences:  A287898 A287899 A287900 * A287902 A287903 A287904

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Jun 02 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
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 27 06:18 EST 2022. Contains 350601 sequences. (Running on oeis4.)