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A191399
Triangle read by rows: T(n,k) is the number of dispersed Dyck paths of semilength n having k peak plateaux.
0
1, 1, 2, 3, 5, 1, 8, 2, 13, 7, 21, 14, 34, 35, 1, 55, 68, 3, 89, 149, 14, 144, 282, 36, 233, 576, 114, 1, 377, 1068, 267, 4, 610, 2088, 711, 23, 987, 3810, 1566, 72, 1597, 7229, 3771, 272, 1, 2584, 13024, 7953, 744, 5, 4181, 24179, 17922, 2304, 34, 6765, 43114, 36594, 5780, 125
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