OFFSET
0,3
COMMENTS
Also, T(n,k) is the number of paths in the first quadrant from (0,0) to (n,0), consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0), having k peaks (i.e., UD's). Row n contains 1+floor(n/2) terms. T(n,0) = A001006(n) (the Motzkin numbers). Row sums yield A128720. Sum(k*T(n,k), k>=0) = A106053(n).
Shifting every column k of this triangle k steps upwards yields triangle A091869, but with another offset. - Werner Schulte, Feb 03 2017
FORMULA
G.f. = G = G(t,z) satisfies G = 1+zG+tz^2*G+z^2*G^2 (see explicit expression at the Maple program).
EXAMPLE
Triangle starts:
1;
1;
2,1;
4,2;
9,6,1;
21,16,3;
51,45,12,1;
T(5,2)=3 because we have hHH, HhH and HHh.
MAPLE
G:=((1-z-t*z^2-sqrt((1+z-t*z^2)*(1-3*z-t*z^2)))*1/2)/z^2: Gser:=simplify(series(G, z=0, 17)): for n from 0 to 13 do P[n]:=sort(coeff(Gser, z, n), n=0..13) end do: for n from 0 to 13 do seq(coeff(P[n], t, j), j=0..floor((1/2)*n)) end do; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Sep 03 2007
EXTENSIONS
Keyword tabf added by Michel Marcus, Apr 09 2013
STATUS
approved