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A064645 Table where the entry (n,k) (n >= 0, k >= 0) gives number of Motzkin paths of the length n with the minimum peak width of k. 8
1, 1, 1, 2, 1, 1, 4, 1, 1, 1, 9, 2, 1, 1, 1, 21, 4, 1, 1, 1, 1, 51, 8, 2, 1, 1, 1, 1, 127, 17, 4, 1, 1, 1, 1, 1, 323, 37, 8, 2, 1, 1, 1, 1, 1, 835, 82, 16, 4, 1, 1, 1, 1, 1, 1, 2188, 185, 33, 8, 2, 1, 1, 1, 1, 1, 1, 5798, 423, 69, 16, 4, 1, 1, 1, 1, 1, 1, 1, 15511, 978, 146, 32, 8, 2, 1, 1, 1, 1, 1, 1, 1, 41835, 2283, 312, 65, 16, 4, 1, 1, 1, 1, 1, 1, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..9869 (rows n=0..139 of triangle, flattened).

EXAMPLE

E.g., we have the following nine Motzkin paths of length 4, of which the last 4 have each peak at least of width 1 and the last 2 with each peak at least 2 dashes wide, so M(4,0) = 9, M(4,1) = 4 and M(4,2) = 2.

   /\                                 _       _     __

  /  \   /\/\   __/\   _/\_   /\__   / \_   _/ \   /  \   ____

MAPLE

[seq(A064645(j), j=0..104)]; A064645 := (n) -> Mpw((((trinv(n)-1)*(((1/2)*trinv(n))+1))-n), (n-((trinv(n)*(trinv(n)-1))/2)));

C := (n, k) -> `if`((n <= 0), 0, binomial(n, k));

Mpw := proc(n, m) local i, k; 1+add(add(A001263(i, k)*C(n-(m*k), 2*i), k=1..i), i=0..floor(n/2)); end;

MATHEMATICA

trinv[n_] := Floor[(1 + Sqrt[8 n + 1])/2];

CC[n_, k_] := If[n <= 0, 0, Binomial[n, k]];

a[n_] := Mpw[(((trinv[n] - 1)*(((1/2) trinv[n]) + 1)) - n), (n - ((trinv[n] (trinv[n] - 1))/2))];

Mpw[n_, m_] := 1 + Sum[Sum[If[k == 0, 0, Binomial[i - 1, k - 1] Binomial[i, k - 1]/k] CC[n - m*k, 2i], {k, 1, i}], {i, 0, n/2}];

Table[a[n], {n, 0, 104}] (* Jean-Fran├žois Alcover, Mar 06 2016, adapted from Maple *)

CROSSREFS

First row (k=0): Motzkin numbers (A001006), second row (k=1): A004148 (with RNA connection), third row (k=2): A004149, fourth row (k=3): A023421, fifth row (k=4): A023422, sixth row (k=5): A023423. Uses the table A001263(n, k) which gives the Dyck paths (Catalan Mountain Ranges) with exactly k peaks. Maple procedure trinv given at A054425.

Sequence in context: A205553 A178411 A257598 * A008307 A249694 A192005

Adjacent sequences:  A064642 A064643 A064644 * A064646 A064647 A064648

KEYWORD

nonn,tabl

AUTHOR

Antti Karttunen, Oct 03 2001

STATUS

approved

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Last modified March 29 07:21 EDT 2017. Contains 284250 sequences.