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A191390 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths of length n with k horizontal segments. 1
1, 0, 1, 1, 1, 0, 3, 2, 3, 1, 0, 8, 2, 5, 8, 7, 0, 22, 12, 1, 14, 22, 31, 3, 0, 64, 50, 12, 42, 64, 117, 28, 1, 0, 196, 184, 78, 4, 132, 196, 416, 162, 18, 0, 625, 648, 390, 52, 1, 429, 625, 1452, 762, 159, 5, 0, 2055, 2256, 1707, 392, 25, 1430, 2055, 5062, 3225, 1012, 85, 1, 0, 6917, 7868, 6954, 2280, 285, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

A dispersed Dyck paths of length n is a Motzkin paths of length n with no (1,0)-steps at positive heights. A horizontal segment is a maximal sequence of consecutive (1,0)-steps.

Row n has 1 + ceiling(n/3) entries.

Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).

T(2n,0) = A000108(n) (the Catalan numbers); T(2n+1,0) = 0.

T(2n-1,1) = T(2n,1) = A014138(n) (partial sums of Catalan numbers).

Sum_{k>=0} k*T(n,k) = A191391(n).

LINKS

Table of n, a(n) for n=0..74.

FORMULA

G.f.: G(t,z) = (2*(1-z+t*z))/(1-z-t*z+(1-z+t*z)*sqrt(1-4*z^2)).

EXAMPLE

T(5,2)=2 because we have (HH)UD(H) and (H)UD(HH), where U=(1,1), D=(1,-1), H=(1,0) (the horizontal segments are shown between parentheses).

Triangle starts:

  1;

  0,  1;

  1,  1;

  0,  3;

  2,  3,  1;

  0,  8,  2;

  5,  8,  7;

  0, 22, 12,  1;

MAPLE

G := (2*(1-z+t*z))/(1-z-t*z+(1-z+t*z)*sqrt(1-4*z^2)): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 17 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 17 do seq(coeff(P[n], t, k), k = 0 .. ceil((1/3)*n)) end do; # yields sequence in triangular form

CROSSREFS

Cf. A000108, A001405, A014138, A191391.

Sequence in context: A324182 A166592 A103497 * A309698 A085747 A106693

Adjacent sequences:  A191387 A191388 A191389 * A191391 A191392 A191393

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Jun 03 2011

STATUS

approved

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Last modified February 25 04:59 EST 2020. Contains 332217 sequences. (Running on oeis4.)