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A101920 Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k up steps starting at an odd height.

%I A101920 #5 Mar 30 2012 17:36:00
%S A101920 2,5,1,13,8,1,34,42,13,1,89,183,102,19,1,233,717,624,205,26,1,610,
%T A101920 2622,3275,1650,366,34,1,1597,9134,15473,11020,3716,602,43,1,4181,
%U A101920 30691,67684,64553,30520,7483,932,53,1,10946,100284,279106,342867,215481
%N A101920 Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k up steps starting at an odd height.
%C A101920 A Schroeder path of length 2n is a lattice path starting from (0,0), ending at (2n,0), consisting only of steps U=(1,1) (up steps), D=(1,-1) (down steps) and H=(2,0) (level steps) and never going below the x-axis (Schroeder paths are counted by the large Schroeder numbers, A006318). Row sums are the large Schroeder numbers (A006318). Column 0 yields the odd-indexed Fibonacci numbers (A001519).
%F A101920 G.f.=G=G(t, z) satisfies tz(1-z)G^2-(1-3z+tz+z^2)G+1-z=0.
%e A101920 T(3,1)=8 because we have HUU'DD, UDUU'DD, UU'DDH, UU'DDUD, UHU'DD, UU'DHD, UU'HDD and UU'UDDD, the up steps starting at odd heights being shown with the prime sign.
%e A101920 Triangle begins:
%e A101920 2;
%e A101920 5,1;
%e A101920 13,8,1;
%e A101920 34,42,13,1;
%e A101920 89,183,102,19,1;
%p A101920 G := 1/2/(-t*z+t*z^2)*(-1+3*z-t*z-z^2+sqrt(1-6*z-2*t*z+11*z^2+2*t*z^2-6*z^3+t^2*z^2-2*t*z^3+z^4)): Gser:=simplify(series(G,z=0,13)):for n from 1 to 11 do P[n]:=coeff(Gser,z^n) od: for n from 1 to 11 do seq(coeff(t*P[n],t^k),k=1..n) od; # yields the sequence in triangular form
%Y A101920 Cf. A006318, A001519, A101919.
%K A101920 nonn,tabl
%O A101920 1,1
%A A101920 _Emeric Deutsch_, Dec 20 2004