OFFSET
0,10
COMMENTS
FORMULA
G.f.: G=G(t,z) satisfies z(1+z-tz^2)G^2-(1+z-z^2)(1+z-tz^2)G + 1+z-z^2=0.
The trivariate g.f. G=G(t,s,z), where t marks odd-level peaks, s marks even-level peaks, and z marks semilength, satisfies aG^2 - bG + c = 0, where a = z(1+z-sz^2), b=(1+z-tz^2)(1+z-sz^2), c=1+z-tz^2.
EXAMPLE
T(6,2)=3 because we have U(UD)DUUU(UD)DDD, UUU(UD)DDDU(UD)D, and UUU(UD)DU(UD)DDD (the even-level peaks are shown between parentheses).
Triangle starts:
1;
0;
0, 1;
1, 0;
0, 1, 1;
2, 2, 0;
1, 3, 3, 1;
5, 8, 4, 0;
5, 13, 12, 6, 1;
...
MAPLE
eq := z*(1+z-t*z^2)*G^2-(1+z-z^2)*(1+z-t*z^2)*G+1+z-z^2 = 0: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 16 do P[n] := sort(expand(coeff(Gser, z, n))) end do: for n from 0 to 16 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, Nov 08 2009
STATUS
approved