cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Original entry on oeis.org

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

Views

Author

Andrei Asinowski, Jun 27 2019

Keywords

Comments

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).

Examples

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

Links

Crossrefs

Cf. A325921.

Programs

  • Mathematica
    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 *)

Formula

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

A325925 Number of Motzkin meanders of length n with an even number of humps and an odd number of peaks.

Original entry on oeis.org

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
Offset: 0

Views

Author

Andrei Asinowski, Jul 14 2019

Keywords

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).

Examples

			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}.

Links

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.

Programs

  • 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 *)

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

A325926 Number of Motzkin excursions of length n with an even number of humps and an odd number of peaks.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 8, 26, 76, 212, 568, 1504, 3968, 10526, 28192, 76398, 209268, 578396, 1609376, 4499336, 12620080, 35482718, 99958776, 282107702, 797637908, 2259545652, 6413273704, 18238099464, 51963195440, 148315593178, 424034498656, 1214186436154
Offset: 0

Views

Author

Andrei Asinowski, Jul 14 2019

Keywords

Comments

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).

Examples

			For n=5, the a(5)=2 paths are UDUHD and UHDUD (2 humps, 1 peak).
For n=6, we have a(6)=8 paths: 6 paths obtained by a permutation of {UD, UHD, H}, and 2 paths obtained by a permutation of {UD, UHHD}.

Links

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: Meanders, #humps=EVEN, #peaks=ODD.
A325926 (this sequence): Excursions, #humps=EVEN, #peaks=ODD.
A325927: Meanders, #humps=ODD, #peaks=ODD.
A325928: Excursions, #humps=ODD, #peaks=ODD.

Programs

  • Mathematica
    CoefficientList[Series[(1/(8*(1 - x)*x^2))* (-Sqrt[(1 - 3*x)*(1 - x)^2*(1 + x)] + Sqrt[(1 - 2*x)*(1 - x)^3*(1 + x + 2*x^2)] - 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)]), {x, 0, 20}], x] (* Vaclav Kotesovec, Aug 09 2019 *)

Formula

G.f.: ( -sqrt((1-t)^2*(1+t)*(1-3*t)) + sqrt((1-2*t)*(1+t+2*t^2)*(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, Aug 09 2019

A325927 Number of Motzkin meanders of length n with an odd number of humps and an odd number of peaks.

Original entry on oeis.org

0, 0, 1, 4, 13, 38, 105, 280, 737, 1942, 5183, 14100, 39151, 110642, 316751, 914248, 2650655, 7701562, 22400559, 65203428, 189970159, 554165922, 1619018259, 4737859512, 13887657307, 40769959314, 119849273449, 352716050428, 1039027117929
Offset: 0

Views

Author

Andrei Asinowski, Aug 10 2019

Keywords

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).

Examples

			For n=3, the a(3)=4 paths are UDH, UDU, UUD, HUD (1 hump, 1 peak).

Links

Crossrefs

Motzkin meanders and excursions with parity 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: Meanders, #humps=EVEN, #peaks=ODD.
A325926: Excursions, #humps=EVEN, #peaks=ODD.
A325927 (this sequence): Meanders, #humps=ODD, #peaks=ODD.
A325928: Excursions, #humps=ODD, #peaks=ODD.

Programs

  • PARI
    seq(n)={my(t='x + O('x*'x^n)); Vec(( 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), -n)} \\ Andrew Howroyd, Aug 12 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).

A325928 Number of Motzkin excursions of length n with an odd number of humps and an odd number of peaks.

Original entry on oeis.org

0, 0, 1, 2, 4, 8, 17, 36, 83, 202, 519, 1382, 3766, 10352, 28551, 78756, 217224, 599542, 1657983, 4598766, 12803044, 35785664, 100412731, 282753476, 798690091, 2262087814, 6421507153, 18265543282, 52047980674, 148554917816, 424656556001, 1215691192244
Offset: 0

Views

Author

Andrei Asinowski, Aug 10 2019

Keywords

Comments

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).

Examples

			For n=4, the a(4)=4 paths are UDHH, HUDH, HHUD, and UUDD (1 hump, 1 peak).

Links

Crossrefs

Motzkin meanders and excursions with parity 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: Meanders, #humps=EVEN, #peaks=ODD.
A325926: Excursions, #humps=EVEN, #peaks=ODD.
A325927: Meanders, #humps=ODD, #peaks=ODD.
A325928 (this sequence): Excursions, #humps=ODD, #peaks=ODD.

Programs

  • PARI
    seq(n)={my(t='x + O('x*'x^n)); Vec(-1/2 + ( -sqrt((1-t)^2*(1+t)*(1-3*t)) + sqrt((1-2*t)*(1+t+2*t^2)*(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)), -n)} \\ Andrew Howroyd, Aug 12 2019

Formula

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

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