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 A325921 Number of Motzkin meanders of length n with an even number of humps and an even number of peaks. 7
 1, 2, 4, 8, 17, 38, 92, 239, 653, 1832, 5192, 14726, 41683, 117822, 333312, 945952, 2698117, 7740920, 22337788, 64788768, 188683267, 551179370, 1613612996, 4731245903, 13888157307, 40804653640, 119984904744, 353085202434, 1039830559085, 3064566227434 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 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 Alois P. Heinz, Table of n, a(n) for n = 0..2100 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.: ( (-1+4*t-3*t^2+sqrt(-3*t^4+4*t^3+2*t^2-4*t+1))/(3*t^2-4*t+1) + (-1+4*t-5*t^2+2*t^3+sqrt(4*t^6-12*t^5+13*t^4-8*t^3+6*t^2-4*t+1))/(-2*t^3+5*t^2-4*t+1) + (-1+4*t-5*t^2+sqrt(5*t^4-4*t^3+6*t^2-4*t+1))/(5*t^2-4*t+1) + (-1+4*t-3*t^2-2*t^3+sqrt(4*t^6+4*t^5-11*t^4+8*t^3+2*t^2-4*t+1))/(2*t^3+3*t^2-4*t+1) ) / (8*t). a(n) ~ 3^(n + 1/2) / (4*sqrt(Pi*n)). - Vaclav Kotesovec, Jul 03 2019 a(n) + A325923(n) = A307575(n). - R. J. Mathar, Jan 25 2023 a(n) + A325925(n) = A307555(n). - R. J. Mathar, Jan 25 2023 EXAMPLE For n=0, 1, 2, 3 there are 2^n paths: all the paths without D (0 humps, 0 peaks). For example, for n=3: UUU, UUH, UHU, UHH, HUU, HUH, HHU, HHH. For n=4, the "extra" path is UDUD (2 humps, 2 peaks). The smallest example with #(humps) <> #(peaks) is UHDUHD (2 humps, 0 peaks). MAPLE b:= proc(x, y, t, p, h) option remember; `if`(x=0, `if`(p+h=0, 1, 0), `if`(y>0, b(x-1, y-1, 0, irem(p+`if`(t=1, 1, 0), 2), irem(h+ `if`(t=2, 1, 0), 2)), 0)+b(x-1, y, `if`(t>0, 2, 0), p, h)+ b(x-1, y+1, 1, p, h)) end: a:= n-> b(n, 0\$4): seq(a(n), n=0..35); # Alois P. Heinz, Jul 03 2019 MATHEMATICA CoefficientList[Series[(1/(8*x))*((-1 + 4*x - 3*x^2 + Sqrt[(-(-1 + x)^2)* (-1 + 2*x + 3*x^2)])/ (1 - 4*x + 3*x^2) - (-1 + 4*x - 5*x^2 + 2*x^3 + Sqrt[(-1 + x)^3*(-1 + x + 4*x^3)])/((-1 + x)^2* (-1 + 2*x)) + (-1 + 4*x - 5*x^2 + Sqrt[1 - 4*x + 6*x^2 - 4*x^3 + 5*x^4])/ (1 - 4*x + 5*x^2) + (-1 + 4*x - 3*x^2 - 2*x^3 + Sqrt[1 - 4*x + 2*x^2 + 8*x^3 - 11*x^4 + 4*x^5 + 4*x^6])/(1 - 4*x + 3*x^2 + 2*x^3)), {x, 0, 30}], x] (* Vaclav Kotesovec, Jul 03 2019 *) CROSSREFS Cf. A307572, A325922. Sequence in context: A090901 A101516 A118928 * A049312 A132043 A055545 Adjacent sequences: A325918 A325919 A325920 * A325922 A325923 A325924 KEYWORD nonn AUTHOR Andrei Asinowski, Jun 27 2019 STATUS approved

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Last modified December 7 07:48 EST 2023. Contains 367630 sequences. (Running on oeis4.)