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A191310 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths (i.e., Motzkin paths with no (1,0) steps at positive heights) of length n having k up-steps starting at level 0. 1
1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 6, 3, 1, 10, 8, 1, 1, 14, 16, 4, 1, 23, 32, 13, 1, 1, 32, 56, 32, 5, 1, 55, 102, 74, 19, 1, 1, 78, 170, 152, 55, 6, 1, 143, 302, 307, 144, 26, 1, 1, 208, 498, 580, 336, 86, 7, 1, 405, 890, 1102, 748, 251, 34, 1, 1, 602, 1478, 2004, 1564, 652, 126, 8, 1, 1228, 2691, 3714, 3200, 1587, 405, 43, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Row n has 1 + floor(n/2) entries.

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

Sum_{k>=0} k*T(n,k) = A093387(n+1).

LINKS

Table of n, a(n) for n=0..80.

FORMULA

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

EXAMPLE

T(5,2)=3 because we have HUDUD, UDHUD, and UDUDH, where U=(1,1), D=(1,-1), H=(1,0).

Triangle starts:

  1;

  1;

  1,  1;

  1,  2;

  1,  4,  1;

  1,  6,  3;

  1, 10,  8,  1;

  1, 14, 16,  4;

  1, 23, 32, 13,  1;

MAPLE

G := 2/(2-2*z-t*(1-sqrt(1-4*z^2))): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 16 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 16 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form

CROSSREFS

Cf. A001405, A093387.

Sequence in context: A130313 A247073 A124428 * A124845 A191392 A127625

Adjacent sequences:  A191307 A191308 A191309 * A191311 A191312 A191313

KEYWORD

nonn,tabf,changed

AUTHOR

Emeric Deutsch, May 30 2011

STATUS

approved

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Last modified July 27 22:07 EDT 2017. Contains 289866 sequences.