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A132888 Triangle read by rows: T(n,k) is the number of paths in the first quadrant, from (0,0) to (n,0), consisting of steps U=[1,1], D[1,-1], h=(1,0) and H=(2,0), having height k (0<=k<=floor(n/2)).

%I A132888 #3 Mar 30 2012 17:36:15
%S A132888 1,1,2,1,3,3,5,10,1,8,27,5,13,72,23,1,21,183,86,7,34,459,302,40,1,55,
%T A132888 1134,994,185,9,89,2782,3158,778,61,1,144,6786,9742,3027,332,11,233,
%U A132888 16498,29450,11196,1607,86,1,377,40014,87602,39795,7117,535,13
%N A132888 Triangle read by rows: T(n,k) is the number of paths in the first quadrant, from (0,0) to (n,0), consisting of steps U=[1,1], D[1,-1], h=(1,0) and H=(2,0), having height k (0<=k<=floor(n/2)).
%C A132888 Row n has 1+floor(n/2) terms. T(n,0)=A000045(n+1) (the Fibonacci numbers). Row sums yield A128720.
%F A132888 G.f. of column k is g[k]=G[k]-G[k-1], where G[0]:=1/(1-z-z^2), G[k]=1/(1-z-z^2-z^2*G[k-1]).
%e A132888 T(3,1)=3 because we have hUD, UhD and UDh.
%e A132888 Triangle starts:
%e A132888 1;
%e A132888 1;
%e A132888 2,1;
%e A132888 3,3;
%e A132888 5,10,1;
%e A132888 8,27,5;
%e A132888 13,72,23,1;
%p A132888 G[0]:=1/(1-z-z^2): for n to 13 do G[n]:=simplify(1/(1-z-z^2-z^2*G[n-1])) end do: g[0]:=G[0]: for n to 13 do g[n]:=simplify(G[n]-G[n-1]) end do: for n from 0 to 13 do gser[n]:=series(g[n],z=0,40) end do: for n from 0 to 13 do seq(coeff(gser[k],z,n),k=0..floor((1/2)*n)) end do;
%Y A132888 Cf. A000045, A128720.
%K A132888 nonn,tabf
%O A132888 0,3
%A A132888 _Emeric Deutsch_, Sep 05 2007