OFFSET
0,5
COMMENTS
LINKS
Alois P. Heinz, Rows n = 0..200, flattened
FORMULA
G.f.: G(t,z) satisfies zG^2-(1-z+tz-3tz^2+3z^2-z^3-t^2z^3+2tz^3)G+1-z+z^2+tz-tz^2=0.
EXAMPLE
T(4,1) = 7 because we have (UU)DDUDUD, UD(UU)DDUD, UDUD(UU)DD, (UU)DUDDUD,
UD(UU)DUDD, (UU)DUDUDD and (UU)DUUDDD, where U=(1,1), D=(1,-1) (the ascents of length 2 starting at an even level are shown between parentheses; note that the last path has an ascent of length 2 that starts at an odd level).
Triangle starts:
1;
1;
1, 1;
2, 3;
6, 7, 1;
18, 19, 5;
54, 59, 18, 1;
MAPLE
G:= 1/2/z*(3*z^2+2*z^3*t+1-z^3*t^2-3*z^2*t-z^3+t*z-z -sqrt(1+20*z^3*t-18*z^5*t^2+15*z^4*t^2+18*z^5*t+6*z^5*t^3-2*z^4*t^3-12*z^2*t -12*z^3 -6*z-24*z^4*t-8*z^3*t^2+z^6-6*z^5+11*z^4 +z^2*t^2+6*z^6*t^2 -4*z^6*t^3 -4*z^6*t+z^6*t^4+2*t*z +11*z^2)): Gser:=simplify(series(G, z=0, 17)): P[0]:=1: for n from 1 to 14 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 14 do seq(coeff(t*P[n], t^j), j=1..1+floor(n/2)) od; # yields sequence in triangular form
# second Maple program:
b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0, `if`(x=0,
`if`(t=2, z, 1), expand(b(x-1, y-1, min(3, t+1))+
`if`(t=2 and irem(y, 2)=0, z, 1)*b(x-1, y+1, 0))))
end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0$2)):
seq(T(n), n=0..14); # Alois P. Heinz, Mar 12 2014
MATHEMATICA
b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x==0, If[t==2, z, 1], Expand[ b[x-1, y-1, Min[3, t+1]] + If[t==2 && Mod[y, 2]==0, z, 1]*b[x-1, y+1, 0]]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 0]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Mar 31 2015, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Nov 29 2005
STATUS
approved