A307557 -id:A307557 - cp's OEIS frontend

A325922 Number of Motzkin excursions of length n with an even number of humps and an even number of peaks.

1, 1, 1, 1, 2, 4, 11, 31, 86, 230, 608, 1588, 4151, 10925, 29083, 78373, 213702, 588366, 1631906, 4550346, 12736029, 35746763, 100561622, 283486702, 800798659, 2266802139, 6429960961, 18276530005, 52051825058, 148520257620, 424507695627

Offset: 0

Andrei Asinowski, Jun 27 2019

nonn

A Motzkin excursion is a lattice path with steps from the set {D=-1, H=0, U=1} that starts at (0,0), never goes below the x-axis, and terminates at the altitude 0.
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).

			For n=3 the a(5)=4 paths are HHHHH, UDUDH, UDHUD, HUDUD.

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).

Cf. A325921.

CoefficientList[Series[(4 (1 - 2 x + 2 x^2) - Sqrt[(1 - 2 x - 3 x^2) (1 - x)^2] - Sqrt[(1 - x - 4 x^3) (1 - x)^3] - Sqrt[(1 + x^2) (1 - 4 x + 5  x^2)] - Sqrt[(1 - 2 x) (1 - 2 x - x^2) (1 - x^2 + 2 x^3)]) / (8  x^2 (1 - x)), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 30 2019 *)

G.f.: (4*(1-2*t+2*t^2) - sqrt((1-2*t-3*t^2)*(1-t)^2) - sqrt((1-t-4*t^3)*(1-t)^3) - sqrt((1+t^2)*(1-4*t+5*t^2)) - sqrt((1-2*t)*(1-2*t-t^2)*(1-t^2+2*t^3)) ) / (8*t^2*(1-t)).
a(n) ~ 3^(n + 3/2) / (8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jul 03 2019
conjecture: a(n)+A325924(n) = A307557(n). - R. J. Mathar, Jan 25 2023

A307555 Number of Motzkin meanders of length n with an even number of humps.

Original entry on oeis.org

1, 2, 4, 8, 17, 40, 106, 307, 927, 2818, 8480, 25142, 73555, 213204, 615074, 1773036, 5121195, 14843518, 43190084, 126112096, 369264395, 1083378784, 3182684838, 9357797643, 27529874201, 81028448678, 238599098824, 702932296258, 2072003987285, 6111009331876

Offset: 0

Cyril Banderier, Apr 14 2019

nonn

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).

			For n = 3, the a(3) = 8 paths are HHH, HHU, HUH, HUU, UHH, UHU, UUU.
For n=5, there are a(5) = 40 paths: 32 paths with no humps, {H, U}^5; and 8 paths with two humps, HUDUD, UDHUD, UDUDH, UDUDU, UDUHD, UDUUD, UHDUD, UUDUD.

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).

Cf. A307557.

a:=gfun[rectoproc]({(15*n^2+45*n+30)*u(n)+(-92*n^2-532*n-696)*u(n+1)+(62*n^2+426*n+672)*u(n+2)+(-32*n^2-264*n-524)*u(n+3)+(-20*n^2-192*n-428)*u(n+4)+(84*n^2+988*n+2848)*u(n+5)+(-70*n^2-906*n-2864)*u(n+6)+(24*n^2+336*n+1140)*u(n+7)+(-3*n^2-45*n-162)*u(n+8), u(0) = 1, u(1) = 2, u(2) = 4, u(3) = 8, u(4) = 17, u(5) = 40, u(6) = 106, u(7) = 307},u(n),remember):
seq(a(n), n=0..30);

G.f.: (sqrt((-t^2+1)/(3*t^2-4*t+1))+sqrt((t^2+1)/(5*t^2-4*t+1))-2)/(4*t).

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, Jan 25 2023

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A325922 Number of Motzkin excursions of length n with an even number of humps and an even number of peaks.

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