The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 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 Adjacent sequences: A191786 A191787 A191788 * A191790 A191791 A191792 KEYWORD nonn AUTHOR Emeric Deutsch, Jun 18 2011 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

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