A191399 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths of semilength n having k peak plateaux.

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).

Links

Table of n, a(n) for n=0..59.

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

Cf. A000045, A001405, A191319.

Sequence in context: A249070 A249069 A191308 * A191316 A191387 A191793

Adjacent sequences: A191396 A191397 A191398 * A191400 A191401 A191402

Keyword

nonn,tabf

Author

Emeric Deutsch, Jun 05 2011

Status

approved