OFFSET
0,3
COMMENTS
A dispersed Dyck paths of semilength n is a Motzkin path of length n with no (1,0)-steps at positive heights. A peak plateau is a run of consecutive peaks that is preceded by an upstep and followed by a down step; a peak consists of an upstep followed by a downstep.
Row n has 1+floor(n/4) entries.
Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).
T(n,0) = A000045(n+1) (Fibonacci numbers).
Sum_{k>=0} k*T(n,k) = A191319(n-2).
FORMULA
G.f.: G=G(t,z) satisfies (t*z^4-z^4-2*z^3+z^2+2*z-1)*G*(1+z*G)+1-z^2=0.
EXAMPLE
T(8,2)=1 because we have (UUDD)(UUDD), where U=(1,1) and D=(1,-1) (the peak plateaux are shown between parentheses).
Triangle starts:
1;
1;
2;
3;
5, 1;
8, 2;
13, 7;
21, 14;
34, 35, 1;
MAPLE
eq := (t*z^4-z^4-2*z^3+z^2+2*z-1)*G*(1+z*G)+1-z^2 = 0: g := RootOf(eq, G): Gser := simplify(series(g, z = 0, 23)): for n from 0 to 19 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 19 do seq(coeff(P[n], t, k), k = 0 .. floor((1/4)*n)) end do; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Jun 05 2011
STATUS
approved