The OEIS is supported by the many generous donors to the OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A325925 Number of Motzkin meanders of length n with an even number of humps and an odd number of peaks. 4
 0, 0, 0, 0, 0, 2, 14, 68, 274, 986, 3288, 10416, 31872, 95382, 281762, 827084, 2423078, 7102598, 20852296, 61323328, 180581128, 532199414, 1569071842, 4626551740, 13641716894, 40223795038, 118614194080, 349847093824, 1032173428200 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS A Motzkin meander is a lattice path with steps from the set {D=-1, H=0, U=1} that starts at (0,0), and never goes below the x-axis. A peak is an occurrence of the pattern UD. A hump is an occurrence of the pattern UHH...HD (the number of Hs in the pattern is not fixed, and can be 0). LINKS Andrei Asinowski, Axel Bacher, Cyril Banderier, Bernhard Gittenberger, Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata, Algorithmica (2019). FORMULA G.f.: ( sqrt((1+t)/(1-3*t)) - sqrt((1+t+2*t^2)/((1-2*t)*(1-t))) + sqrt((1+t^2)/(1-4*t+5*t^2))  - sqrt((1-t^2+2*t^3)/((1-2*t)*(1-t^2-2*t))) ) / (8*t). a(n) ~ 3^(n + 1/2) / (4*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 09 2019 EXAMPLE For n=5, the a(5)=2 paths are UDUHD and UHDUD (2 humps, 1 peak). For n=6, we have a(6)=14 paths: 6 paths obtained by a permutation of {UD, UHD, H}, 6 paths obtained by a permutation of {UD, UHD, U}, and 2 paths obtained by a permutation of {UD, UHHD}. MATHEMATICA CoefficientList[Series[(Sqrt[(1 + x)/(1 - 3*x)] - Sqrt[(1 + x + 2*x^2)/((1 - 2*x)*(1 - x))] + Sqrt[(1 + x^2)/(1 - 4*x + 5*x^2)] - Sqrt[(1 - x^2 + 2*x^3)/((1 - 2*x)*(1 - 2*x - x^2))])/(8*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Aug 09 2019 *) CROSSREFS Motzkin meanders and excursions with restrictions on the number of humps and peaks: A325921: Meanders, #humps=EVEN, #peaks=EVEN. A325922: Excursions, #humps=EVEN, #peaks=EVEN. A325923: Meanders, #humps=ODD, #peaks=EVEN. A325924: Excursions, #humps=ODD, #peaks=EVEN. A325925 (this sequence): Meanders, #humps=EVEN, #peaks=ODD. A325926: Excursions, #humps=EVEN, #peaks=ODD. A325927: Meanders, #humps=ODD, #peaks=ODD. A325928: Excursions, #humps=ODD, #peaks=ODD. Sequence in context: A109869 A197777 A197608 * A084132 A271235 A084770 Adjacent sequences:  A325922 A325923 A325924 * A325926 A325927 A325928 KEYWORD nonn AUTHOR Andrei Asinowski, Jul 14 2019 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 19 13:24 EDT 2022. Contains 353833 sequences. (Running on oeis4.)