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%I #5 Dec 29 2023 12:59:55
%S 1,1,2,3,1,6,3,11,10,23,26,2,47,70,10,102,176,45,221,449,160,5,493,
%T 1121,539,35,1105,2817,1680,196,2516,7031,5082,868,14,5763,17604,
%U 14856,3486,126,13328,43996,42660,12810,840,30995,110147,120338,44640,4410,42
%N Triangle read by rows: T(n,k) is number of Motzkin paths of length n having k UH's, where U=(1,1), H=(1,0) (0<=k<=floor(n/3)).
%C Row n contains 1+floor(n/3) terms. Row sums are the Motzkin numbers (A001006). Column 0 yields A090344. Sum(k*T(n,k),k=0..floor(n/3))=A014531(n-2).
%F G.f.=[1-z-sqrt(1-2z-3z^2-4tz^3+4z^3)]/[2(1-z+tz)z^2].
%e T(4,1)=3 because we have H(UH)D, (UH)DH and (UH)HD, where U=(1,1), H=(1,0), D=(1,-1) (the UH's are shown between parentheses).
%e Triangle begins:
%e 1;
%e 1;
%e 2;
%e 3,1;
%e 6,3;
%e 11,10;
%e 23,26,2;
%e 47,70,10;
%p G:=(1-z-sqrt(1-2*z-3*z^2-4*z^3*t+4*z^3))/2/z^2/(1-z+t*z): Gser:=simplify(series(G,z=0,20)): P[0]:=1: for n from 1 to 16 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 16 do seq(coeff(t*P[n],t^j),j=1..1+floor(n/3)) od; # yields sequence in triangular form
%Y Cf. A001006, A090344, A014531.
%K nonn,tabf
%O 0,3
%A _Emeric Deutsch_, Dec 09 2005
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