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A191789 Number of length n left factors of Dyck paths having no base pyramids. 2
1, 1, 1, 2, 3, 6, 11, 21, 40, 76, 146, 279, 539, 1036, 2011, 3883, 7566, 14662, 28654, 55692, 109098, 212564, 417210, 814568, 1601366, 3132078, 6165732, 12077905, 23803827, 46691096, 92113651, 180893091, 357183430, 702169718, 1387539542, 2730236900, 5398831722, 10632066436 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

A base pyramid is a factor of the form U^j D^j (j>0), starting at the horizontal axis (here U=(1,1) and D=(1,-1)).

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

FORMULA

a(n) = A191788(n,0).

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

a(n) ~ 3*2^(n+1/2)/(5*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 21 2014

EXAMPLE

a(4)=3 because we have UUDU, UUUD, and UUUU; each of the paths (UD)(UD), (UD)UU, and (UUDD) has at least one base pyramid (shown between parentheses).

MAPLE

c := ((1-sqrt(1-4*z^2))*1/2)/z^2: G := (1-z^2)*c/((1-z*c)*(1+z^4*c^2)): Gser := series(G, z = 0, 40): seq(coeff(Gser, z, n), n = 0 .. 37);

MATHEMATICA

With[{c=(1-Sqrt[1-4x^2])/(2x^2)}, CoefficientList[Series[(1-x^2)c/ ((1-x c ) (1+x^4 c^2)), {x, 0, 40}], x]] (* Harvey P. Dale, Jun 19 2011 *)

PROG

(PARI) x='x+O('x^50); Vec( 2*(1-x^2)*(1-sqrt(1-4*x^2))/(x*(2*x-1+sqrt(1-4*x^2))*(3-2*x^2-sqrt(1-4*x^2))) ) \\ G. C. Greubel, Mar 27 2017

CROSSREFS

Cf. A191788.

Sequence in context: A316796 A079116 A109222 * A006861 A052956 A298118

Adjacent sequences:  A191786 A191787 A191788 * A191790 A191791 A191792

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Jun 18 2011

STATUS

approved

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Last modified March 21 22:19 EDT 2019. Contains 321382 sequences. (Running on oeis4.)