The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A114500 Number of Dyck paths of semilength n having no UUUDDD's starting at level zero; here U=(1,1), D=(1,-1). Also number of Dyck paths of semilength n having no UUDUDD's starting at level zero. 2
 1, 1, 2, 4, 12, 37, 119, 390, 1307, 4460, 15452, 54207, 192170, 687386, 2477810, 8992007, 32825653, 120460613, 444125661, 1644324767, 6111002752, 22789116600, 85251100275, 319826371389, 1203008722282, 4536009027311, 17141555233270 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Column 0 of A114499. LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 FORMULA G.f.: 1/(1 - z*C + z^3), where C = (1-sqrt(1-4*z))/(2*z) is the Catalan function. a(n) ~ 4^(n+5)/(1089*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014 EXAMPLE a(4)=12 because among the 14 Dyck paths of semilength 4 only UDUUUDDD and UUUDDDUD contain UUUDDD starting at level 0. MAPLE C:=(1-sqrt(1-4*z))/2/z: G:=1/(1-z*C+z^3): Gser:=series(G, z=0, 35): 1, seq(coeff(Gser, z^n), n=1..30); MATHEMATICA CoefficientList[Series[1/(1-x*(1-Sqrt[1-4*x])/2/x+x^3), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *) PROG (PARI) x='x+O('x^50); Vec(1/(1-x*(1-sqrt(1-4*x))/(2*x*(1+x^2))) \\ G. C. Greubel, Mar 17 2017 CROSSREFS Cf. A114499. Sequence in context: A255432 A275539 A193049 * A148212 A139627 A149844 Adjacent sequences:  A114497 A114498 A114499 * A114501 A114502 A114503 KEYWORD nonn AUTHOR Emeric Deutsch, Dec 04 2005 STATUS approved

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

Last modified December 4 02:40 EST 2021. Contains 349469 sequences. (Running on oeis4.)