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Triangle read by rows: T(n,k) is the number of Motzkin paths of length n and having k peaks at even height.
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%I #4 Mar 30 2012 17:35:59

%S 1,1,2,4,8,1,17,4,38,12,1,88,34,5,209,95,18,1,506,264,59,6,1244,731,

%T 187,25,1,3097,2020,582,92,7,7791,5578,1786,322,33,1,19773,15404,5420,

%U 1096,134,8,50563,42558,16308,3652,510,42,1,130149,117652,48744,11960,1872

%N Triangle read by rows: T(n,k) is the number of Motzkin paths of length n and having k peaks at even height.

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

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

%e Triangle begins:

%e 1;

%e 1;

%e 2;

%e 4;

%e 8,1;

%e 17,4;

%e 38,12,1;

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

%e T(5,1)=4 counts HU(UD)D, U(UD)DH, UH(UD)D and U(UD)HD, where U=(1,1), H=(1,0), D=(1,-1) (the peaks at even height are shown between parentheses).

%Y Cf. A001006.

%K nonn,tabf

%O 0,3

%A _Emeric Deutsch_, Sep 03 2004