cp's OEIS Frontend

A101895 Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k peaks at even height.

%I A101895 #5 Mar 30 2012 17:36:00
%S A101895 2,5,1,15,6,1,51,30,8,1,188,144,51,10,1,731,685,300,77,12,1,2950,3258,
%T A101895 1695,532,108,14,1,12235,15533,9348,3455,854,144,16,1,51822,74280,
%U A101895 50729,21538,6245,1280,185,18,1,223191,356283,272128,130375,43278,10387,1824
%N A101895 Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k peaks at even height.
%C A101895 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 A101895 G.f.=G=G(t, z) satisfies z(1-z)G^2-(1-z)(1-tz)G+1-tz=0.
%e A101895 T(3,1)=6 because we have HU(UD)D, U(UD)DH, UH(UD)D, U(UD)HD, UDU(UD)D and
%e A101895 U(UD)DUD, the peaks at even height being shown between parentheses.
%e A101895 Triangle begins:
%e A101895 2;
%e A101895 5,1;
%e A101895 15,6,1;
%e A101895 51,30,8,1;
%e A101895 188,144,51,10,1;
%p A101895 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 A101895 Cf. A006318, A007317, A026376, A101894.
%K A101895 nonn,tabl
%O A101895 1,1
%A A101895 _Emeric Deutsch_, Dec 20 2004