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 A307572 Number of Motzkin meanders of length n with an odd number of humps. 2
 0, 0, 1, 5, 18, 56, 161, 443, 1196, 3228, 8823, 24579, 69810, 201380, 586843, 1719081, 5044584, 14800352, 43384747, 127076015, 372100654, 1089864344, 3194496987, 9372984609, 27532712140, 80966582548, 238342592353, 702222958797, 2070454005078, 6108341367004 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 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 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..1000 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^2)/(1-4*t+3*t^2)) - sqrt((1+t^2)/(1-4*t+5*t^2))) / (4*t). Conjecture: D-finite with recurrence -3*(n+1)*(n-2)*a(n) +12*(2*n^2-4*n-1)*a(n-1) +2*(-35*n^2+107*n-48)*a(n-2) +4*(21*n^2-89*n+80)*a(n-3) +4*(-5*n^2+32*n-43)*a(n-4) +4*(-8*n^2+62*n-115)*a(n-5) +2*(31*n^2-283*n+616)*a(n-6) -4*(23*n-97)*(n-6)*a(n-7) +15*(n-6)*(n-7)*a(n-8)=0. - R. J. Mathar, May 06 2020 a(n) ~ 3^(n + 1/2) / (2*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 08 2023 EXAMPLE For n = 3 the a(3) = 5 paths are UDH, HUD, UHD, UUD, UDU. MAPLE b:= proc(x, y, t, c) option remember; `if`(y<0, 0, `if`(x=0, c, b(x-1, y-1, 0, irem(c+t, 2))+b(x-1, y, t, c)+b(x-1, y+1, 1, c))) end: a:= n-> b(n, 0\$3): seq(a(n), n=0..35); # Alois P. Heinz, Apr 16 2019 MATHEMATICA b[x_, y_, t_, c_] := b[x, y, t, c] = If[y<0, 0, If[x==0, c, b[x-1, y-1, 0, Mod[c+t, 2]] + b[x-1, y, t, c] + b[x-1, y+1, 1, c]]]; a[n_] := b[n, 0, 0, 0]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Apr 29 2019, after Alois P. Heinz *) CROSSREFS Cf. A001006. Sequence in context: A001793 A325919 A317849 * A325923 A335720 A093374 Adjacent sequences: A307569 A307570 A307571 * A307573 A307574 A307575 KEYWORD nonn AUTHOR Andrei Asinowski, Apr 15 2019 STATUS approved

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Last modified August 13 14:39 EDT 2024. Contains 375142 sequences. (Running on oeis4.)