OFFSET
0,6
COMMENTS
LINKS
FORMULA
The g.f. G=G(t,z) satisfies z(1-z)G^2 - (1-z+z^2-tz^2)G+1-z = 0 (for the explicit form of G see the Maple program).
The trivariate g.f. g=g(x,y,z) of Dyck paths with respect to number of peak plateaux, number of peaks in the peak plateaux and semilength, marked, by x, y and z, respectively satisfies g=1+zg[g+xyz/(1-yz)-z/(1-z)].
T(n,k) = Sum_{r=1..n} Narayana(n-r,k)*binomial(2n-r-k,r-k) where Narayana(n,k) := binomial(n,k)*binomial(n,k-1)/n is the Narayana number A001263. - David Callan, Oct 31 2008
EXAMPLE
T(3,1)=4 because we have UD(UUDD), (UUDD)UD, (UUDUDD) and U(UUDD)D (the peak plateaux are shown between parentheses).
The triangle starts:
1;
1;
1,1;
1,4;
1,12,1;
1,33,8;
1,88,42,1;
MAPLE
C:=proc(z) options operator, arrow: (1/2-(1/2)*sqrt(1-4*z))/z end proc: G:=(1-z)*C(z*(1-z)^2/(1-z+z^2-t*z^2)^2)/(1-z+z^2-t*z^2): Gser:=simplify(series(G, z= 0, 17)): for n from 0 to 14 do P[n]:=sort(coeff(Gser, z, n)) end do: for n from 0 to 14 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, Oct 10 2008
STATUS
approved