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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. 9

%I

%S 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 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 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 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 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 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 /\ _ _ __

%e / \ /\/\ __/\ _/\_ /\__ / \_ _/ \ / \ ____

%e The array starts:

%e 1 1 1 1 1 1 1 1 1 1 1

%e 1 1 1 1 1 1 1 1 1 1 1

%e 2 1 1 1 1 1 1 1 1 1 1

%e 4 2 1 1 1 1 1 1 1 1 1

%e 9 4 2 1 1 1 1 1 1 1 1

%e 21 8 4 2 1 1 1 1 1 1 1

%e 51 17 8 4 2 1 1 1 1 1 1

%e 127 37 16 8 4 2 1 1 1 1 1

%e 323 82 33 16 8 4 2 1 1 1 1

%e 835 185 69 32 16 8 4 2 1 1 1

%e 2188 423 146 65 32 16 8 4 2 1 1

%e 5798 978 312 133 64 32 16 8 4 2 1

%e 15511 2283 673 274 129 64 32 16 8 4 2

%p # trinv() given in A054425

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

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

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

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

%t 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 Table[a[n], {n, 0, 104}] (* _Jean-Fran├žois Alcover_, Mar 06 2016, adapted from Maple *)

%Y 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 nonn,tabl

%O 0,4

%A _Antti Karttunen_, Oct 03 2001

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Last modified October 16 20:34 EDT 2018. Contains 316275 sequences. (Running on oeis4.)