A101894 -id:A101894 - 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.

2, 5, 1, 15, 6, 1, 51, 30, 8, 1, 188, 144, 51, 10, 1, 731, 685, 300, 77, 12, 1, 2950, 3258, 1695, 532, 108, 14, 1, 12235, 15533, 9348, 3455, 854, 144, 16, 1, 51822, 74280, 50729, 21538, 6245, 1280, 185, 18, 1, 223191, 356283, 272128, 130375, 43278, 10387, 1824

Offset: 1

Emeric Deutsch, Dec 20 2004

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.

			T(3,1)=6 because we have HU(UD)D, U(UD)DH, UH(UD)D, U(UD)HD, UDU(UD)D and
U(UD)DUD, the peaks at even height being shown between parentheses.
Triangle begins:
2;
5,1;
15,6,1;
51,30,8,1;
188,144,51,10,1;

Cf. A006318, A007317, A026376, A101894.

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

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

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.

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