OFFSET
0,8
COMMENTS
FORMULA
The trivariate g.f. G=G(t,s,z), where z marks semilength, t marks odd-level peaks and s marks even-level peaks, satisfies G = 1 + tzG + sz^2*G + s^2*z^3*HG, where H=G(s,t,z) (interchanging t and s and eliminating H, one obtains G(t,s,z); see the Maple program).
EXAMPLE
T(4,2)=3 because we have (UD)(UD)UUDD, (UD)UUDD(UD), and UUDD(UD)(UD) (the odd level peaks are shown between parentheses).
Triangle starts:
1;
0,1;
1,0,1;
1,2,0,1;
2,2,3,0,1;
3,6,3,4,0,1.
MAPLE
p1 := -G+1+t*z*G+s*z^2*G+s^2*z^3*H*G: p2 := subs({t = s, s = t, G = H, H = G}, p1): r := resultant(p1, p2, H): G := RootOf(subs(s = 1, r), G): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 12 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 12 do seq(coeff(P[n], t, j), j = 0 .. n) end do; yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Oct 12 2009
STATUS
approved