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Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k peaks at even height.
1

%I #5 Mar 30 2012 17:36:00

%S 2,5,1,15,6,1,51,30,8,1,188,144,51,10,1,731,685,300,77,12,1,2950,3258,

%T 1695,532,108,14,1,12235,15533,9348,3455,854,144,16,1,51822,74280,

%U 50729,21538,6245,1280,185,18,1,223191,356283,272128,130375,43278,10387,1824

%N Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k peaks at even height.

%C 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 A007317. Column 1 yields A026376.

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

%e T(3,1)=6 because we have HU(UD)D, U(UD)DH, UH(UD)D, U(UD)HD, UDU(UD)D and

%e U(UD)DUD, the peaks at even height being shown between parentheses.

%e Triangle begins:

%e 2;

%e 5,1;

%e 15,6,1;

%e 51,30,8,1;

%e 188,144,51,10,1;

%p G := 1/2/(-z+z^2)*(-1+t*z+z-t*z^2+sqrt(1-2*t*z-6*z+8*t*z^2+t^2*z^2-2*t^2*z^3+5*z^2-6*t*z^3+t^2*z^4)): Gser:=simplify(series(G,z=0,14)): for n from 1 to 12 do P[n]:=coeff(Gser,z^n) od: for n from 1 to 12 do seq(coeff(t*P[n],t^k),k=1..n) od; # yields the sequence in triangular form

%Y Cf. A006318, A007317, A026376, A101894.

%K nonn,tabl

%O 1,1

%A _Emeric Deutsch_, Dec 20 2004