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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
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
D-finite with recurrence -94*(n+1)*(n-15)*a(n) +2*(-47*n^2+734*n-2115)*a(n-1) +8*(88*n^2-1311*n+1307)*a(n-2) +2*(235*n^2-3904*n+13597)*a(n-3) +(-1593*n^2+25078*n-72033)*a(n-4) +(-423*n^2+7074*n-25079)*a(n-5) +(1241*n^2-19834*n+66805)*a(n-6) +2*(94*n^2-1585*n+5741)*a(n-7) -4*(n-7)*(117*n-1087)*a(n-8)=0. - R. J. Mathar, Jul 24 2022
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 * A371790 A306575 A006861
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Jun 18 2011
STATUS
approved

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Last modified May 23 10:34 EDT 2024. Contains 372760 sequences. (Running on oeis4.)