0,5

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). Also number of Schroeder paths of length 2n and having k humps. A hump is an up step U followed by 0 or more level steps H followed by a down step D. The T(3,2)=8 Schroeder paths of length 6 and having 2 humps are: H(UD)(UD), (UD)H(UD), (UD)(UD)H, (UD)(UHD), (UD)(UUDD), (UHD)(UD), (UUDD)(UD), and U(UD)(UD)D, the humps being shown between parentheses. Row sums are the large Schroeder numbers (A006318). Column 1 yields the odd-indexed Fibonacci numbers minus 1 (A027941). T(n,n-1)=A034856(n)=binomial(n + 1, 2) + n - 1.

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

T(3,2)=8 because we have HU'DU'D, U'DHU'D, U'DU'DH, U'DU'HD, U'DU'UDD, U'HDU'D, U'UDDU'D, and U'UU'DDD, the up steps starting at an even height being shown with a prime sign..

Triangle begins:

1;

1,1;

14,1;

1,12,8,1;

1,33,42,13,1;

G:=1/2/(-z+z^2)*(-1+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)): P[0]:=1: for n from 1 to 11 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 11 do seq(coeff(t*P[n], t^k), k=1..n+1) od; # yields the sequence in triangular form

Cf. A006318, A027941, A034856, A101920.

Adjacent sequences: A101916 A101917 A101918 this_sequence A101920 A101921 A101922

Sequence in context: A087903 A112500 A008292 this_sequence A055106 A080416 A099759

nonn,tabl

Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 20 2004