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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.

%I A064645 #24 Jul 23 2017 21:56:39
%S A064645 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,
%T A064645 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,
%U A064645 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
%N 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.
%H A064645 Seiichi Manyama, <a href="/A064645/b064645.txt">Table of n, a(n) for n = 0..9869 (rows n=0..139 of triangle, flattened)</a>
%e A064645 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.
%e A064645    /\                                 _       _     __
%e A064645   /  \   /\/\   __/\   _/\_   /\__   / \_   _/ \   /  \   ____
%e A064645 The array starts:
%e A064645       1    1   1   1   1   1   1   1   1   1   1
%e A064645       1    1   1   1   1   1   1   1   1   1   1
%e A064645       2    1   1   1   1   1   1   1   1   1   1
%e A064645       4    2   1   1   1   1   1   1   1   1   1
%e A064645       9    4   2   1   1   1   1   1   1   1   1
%e A064645      21    8   4   2   1   1   1   1   1   1   1
%e A064645      51   17   8   4   2   1   1   1   1   1   1
%e A064645     127   37  16   8   4   2   1   1   1   1   1
%e A064645     323   82  33  16   8   4   2   1   1   1   1
%e A064645     835  185  69  32  16   8   4   2   1   1   1
%e A064645    2188  423 146  65  32  16   8   4   2   1   1
%e A064645    5798  978 312 133  64  32  16   8   4   2   1
%e A064645   15511 2283 673 274 129  64  32  16   8   4   2
%p A064645 # trinv() given in A054425
%p A064645 [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)));
%p A064645 C := (n,k) -> `if`((n <= 0),0,binomial(n,k));
%p A064645 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;
%t A064645 trinv[n_] := Floor[(1 + Sqrt[8 n + 1])/2];
%t A064645 CC[n_, k_] := If[n <= 0, 0, Binomial[n, k]];
%t A064645 a[n_] := Mpw[(((trinv[n] - 1)*(((1/2) trinv[n]) + 1)) - n), (n - ((trinv[n] (trinv[n] - 1))/2))];
%t A064645 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}];
%t A064645 Table[a[n], {n, 0, 104}] (* _Jean-François Alcover_, Mar 06 2016, adapted from Maple *)
%Y A064645 Column k=0: Motzkin numbers (A001006), column k=1: A004148, column k=2: A004149, column k=3: A023421, column k=4: A023422, column k=5: A023423. Uses the table A001263(n, k).
%K A064645 nonn,tabl
%O A064645 0,4
%A A064645 _Antti Karttunen_, Oct 03 2001