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Triangle read by rows: T(n,k) is the number of Motzkin paths of length n having k high humps. (A hump is an upstep followed by 0 or more flatsteps followed by a downstep. A high hump is a hump that starts at a level higher than zero.).
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%I #4 Mar 30 2012 17:35:59

%S 1,1,2,4,8,1,16,5,32,18,1,64,56,7,128,160,34,1,256,432,138,9,512,1120,

%T 500,55,1,1024,2816,1672,275,11,2048,6912,5264,1205,81,1,4096,16640,

%U 15808,4797,481,13,8192,39424,45696,17738,2471,112,1,16384,92160,128000

%N Triangle read by rows: T(n,k) is the number of Motzkin paths of length n having k high humps. (A hump is an upstep followed by 0 or more flatsteps followed by a downstep. A high hump is a hump that starts at a level higher than zero.).

%C Row sums are the Motzkin numbers (A001006).

%F G.f.=G=G(t, z) satisfies tz^2*(1-z)G^2-(1-2*z+tz^2)*G+1-z=0.

%e Triangle begins:

%e 1;

%e 1;

%e 2;

%e 4;

%e 8,1;

%e 16,5;

%e 32,18,1;

%e Row n contains floor(n/2) terms.

%e T(5,1)=5 counts HU(UD)D, U(UD)DH, UH(UD)D, U(UD)HD and U(UHD)D, where U=(1,1), H=(1,0), D=(1,-1) (the high humps are shown between parentheses).

%Y Cf. A001006.

%K nonn,tabf

%O 0,3

%A _Emeric Deutsch_, Sep 02 2004