OFFSET
0,5
COMMENTS
FORMULA
G.f. G=G(s,z) satisfies G = 1 + zG(1 + szG)/(1 - z^2*G^2).
The trivariate g.f. H=H(t,s,z), where t (s) marks odd-length (even-length) ascents satisfies H = 1 + zH(t+szH)/(1-z^2*H^2).
EXAMPLE
T(4,1)=7 because we have UDUD(UU)DD, UD(UU)DDUD, UD(UU)DUDD, (UU)DDUDUD, (UU)DUDDUD, (UU)DUDUDD and (UUUU)DDDD (the even-length ascents are shown between parentheses).
Triangle starts:
1;
1;
1, 1;
2, 3;
5, 7, 2;
12, 20, 10;
30, 61, 36, 5;
MAPLE
eq:=G=1+(1+s*z*G)*z*G/(1-z^2*G^2): G:=RootOf(eq, G): Gser:=simplify(series(G, z =0, 16)): for n from 0 to 13 do P[n]:=sort(expand(coeff(Gser, z, n))) end do: for n from 0 to 13 do seq(coeff(P[n], s, j), j=0..floor((1/2)*n)) end do; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Oct 05 2008
STATUS
approved