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

Author

Emeric Deutsch, Jun 05 2011

Keywords

• nonn
• tabf

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).

Examples

			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;
		

Crossrefs

Cf. A000045, A001405, A191319.

Programs

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

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.