login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A288108 Number T(n,k) of Dyck paths of semilength n such that each level has exactly k peaks or no peaks; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 13
1, 0, 1, 0, 1, 1, 0, 3, 1, 1, 0, 5, 2, 1, 1, 0, 13, 5, 3, 1, 1, 0, 31, 15, 4, 4, 1, 1, 0, 71, 27, 10, 7, 5, 1, 1, 0, 181, 76, 36, 11, 11, 6, 1, 1, 0, 447, 196, 83, 22, 19, 16, 7, 1, 1, 0, 1111, 548, 225, 81, 32, 31, 22, 8, 1, 1, 0, 2799, 1388, 573, 235, 60, 56, 48, 29, 9, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

T(n,k) is defined for all n,k >= 0.  The triangle contains only the terms for k<=n. T(0,k) = 1 and T(n,k) = 0 if k > n > 0.

LINKS

Alois P. Heinz, Rows n = 0..140, flattened

Wikipedia, Counting lattice paths

EXAMPLE

. T(5,2) = 5:                                        /\/\

.                                       /\  /\      /    \

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

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

.

. T(5,3) = 3:

.                                       /\/\/\

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

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

.

Triangle T(n,k) begins:

  1;

  0,   1;

  0,   1,  1;

  0,   3,  1,  1;

  0,   5,  2,  1,  1;

  0,  13,  5,  3,  1,  1;

  0,  31, 15,  4,  4,  1, 1;

  0,  71, 27, 10,  7,  5, 1, 1;

  0, 181, 76, 36, 11, 11, 6, 1, 1;

MAPLE

b:= proc(n, k, j) option remember; `if`(n=j, 1, add(

      b(n-j, k, i)*(binomial(j-1, i-1)+binomial(i, k)

       *binomial(j-1, i-1-k)), i=1..min(j+k, n-j)))

    end:

T:= (n, k)-> b(n, k$2):

seq(seq(T(n, k), k=0..n), n=0..14);

MATHEMATICA

b[n_, k_, j_] := b[n, k, j] = If[n == j, 1, Sum[b[n - j, k, i]*(Binomial[j - 1, i - 1] + Binomial[i, k]*Binomial[j - 1, i - 1 - k]), {i, 1, Min[j + k, n - j]}]];

T[n_, k_] := b[n, k, k];

Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, May 25 2018, translated from Maple *)

CROSSREFS

Columns k=0-10 give: A000007, A281874, A287843, A288110, A288111, A288112, A288113, A288114, A288115, A288116, A288117.

Row sums give A288109.

T(2n,n) gives A156043.

Cf. A000108, A288318.

Sequence in context: A130115 A191582 A130160 * A287822 A162169 A216954

Adjacent sequences:  A288105 A288106 A288107 * A288109 A288110 A288111

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jun 05 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 May 31 00:29 EDT 2020. Contains 334747 sequences. (Running on oeis4.)