A274256 -id:A274256 - cp's OEIS frontend

A274052 Number of factor-free Dyck words with slope 5/2 and length 7n.

1, 3, 13, 94, 810, 7667, 76998, 805560, 8684533, 95800850, 1076159466, 12268026894, 141565916433, 1650395185407, 19409211522550, 229984643863260, 2743097412254490, 32907239462485422, 396793477697214450, 4806417317271974580, 58460150525944945840, 713685698665966837135, 8742060290902752902340

Offset: 0

Michael D. Weiner, Jun 08 2016

nonn

a(n) is the number of lattice paths (allowing only north and east steps) starting at (0,0) and ending at (2n,5n) that stay below the line y = 5/2x and also do not contain a proper subpath of smaller size.

			a(2) = 13 since there are 13 lattice paths (allowing only north and east steps) starting at (0,0) and ending at (4,10) that stay below the line y=5/2x and also do not contain a proper subpath of small size; e.g., EEENNNENNNNNNN is a factor-free Dyck word but ENNEENENNNNNNN contains the factor ENENNNN.

Cyril Banderier and Michael Wallner, Lattice paths of slope 2/5, 2015 Proceedings of the Twelfth Workshop on Analytic Algorithmics and Combinatorics (ANALCO).
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, On rational Dyck paths and the enumeration of factor-free Dyck words, arXiv:1606.02183 [math.CO], 2016.
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, On factor-free Dyck words with half-integer slope, arXiv:1804.11244 [math.CO], 2018.
P. Duchon, On the enumeration and generation of generalized Dyck words, Discrete Mathematics, 225 (2000), 121-135.

Factor-free Dyck words: A005807 (slope 3/2), A274244 (slope 7/2), A274256 (slope 9/2), A274257 (slope 4/3), A274259 (slope 7/3).

Cf. A293946, A322631.

Conjectural o.g.f.: Let E(x) = exp( Sum_{n >= 1} binomial(7*n, 2*n)*x^n/n ). Then A(x) = ( x/series reversion of x*E(x) )^(1/7) = 1 + 3*x + 13*x^2 + 94*x^3 + ... . Equivalently, [x^n]( A(x)^(7*n) ) = binomial(7*n, 2*n) for n = 0,1,2,... . - Peter Bala, Jan 01 2020

A274244 Number of factor-free Dyck words with slope 7/2 and length 9n.

Original entry on oeis.org

1, 4, 34, 494, 8615, 165550, 3380923, 71999763, 1580990725, 35537491360, 813691565184, 18911247654404, 444978958424224, 10579389908116344, 253756528273411250, 6133110915783398175, 149219383150626519874, 3651756292682801022384, 89830021324956206790496, 2219945238901447637080235, 55088272581138888326634644

Offset: 0

Michael D. Weiner, Jun 15 2016

nonn

a(n) is the number of lattice paths (allowing only north and east steps) starting at (0,0) and ending at (2n,7n) that stay below the line y=7/2x and also do not contain a proper subpath of smaller size.

			a(2) = 34 since there are 34 lattice paths (allowing only north and east steps) starting at (0,0) and ending at (4,14) that stay below the line y=7/2x and also do not contain a proper subpath of small size; e.g., EEENNNNENNNNNNNNNN is a factor-free Dyck word but EEENNENNNNNNNNNNNN contains the factor ENNENNNNN.

Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, On rational Dyck paths and the enumeration of factor-free Dyck words, arXiv:1606.02183 [math.CO], 2016.
P. Duchon, On the enumeration and generation of generalized Dyck words, Discrete Mathematics, 225 (2000), 121-135.

Factor-free Dyck words: A005807 (slope 3/2), A274052 (slope 5/2), A274256 (slope 9/2), A274257 (slope 4/3), A274259 (slope 7/3).

Cf. A060941.

Conjectural o.g.f.: Let E(x) = exp( Sum_{n >= 1} binomial(9*n, 2*n)*x^n/n ). Then A(x) = ( x/series reversion of x*E(x) )^(1/9) = 1 + 4*x + 34*x^2 + 494*x^3 + ... . Equivalently, [x^n]( A(x)^(9*n) ) = binomial(9*n, 2*n) for n = 0,1,2,... . - Peter Bala, Jan 01 2020

A274257 Number of factor-free Dyck words with slope 4/3 and length 7n.

Original entry on oeis.org

1, 5, 52, 880, 17856, 399296, 9491008, 235274240, 6014201600, 157387037696, 4195621863424, 113534211297280, 3110485641494528, 86107512380129280, 2404899661362184192, 67680890349732102144, 1917436905101367443456, 54640222663002565640192, 1565130555077611323392000, 45039415225401829826232320

Offset: 0

Michael D. Weiner, Jun 16 2016

nonn

a(n) is the number of lattice paths (allowing only north and east steps) starting at (0,0) and ending at (3n,4n) that stay below the line y=4/3x and also do not contain a proper subpath of smaller size.

			a(2) = 52 since there are 52 lattice paths (allowing only north and east steps) starting at (0,0) and ending at (6,8) that stay below the line y=4/3x and also do not contain a proper subpath of small size; e.g., EENEENENNENNNN is a factor-free Dyck word but ENEEENNENNENNN contains the factor EENNENN.

Daniel Birmajer, Juan B. Gil and Michael D. Weiner, On rational Dyck paths and the enumeration of factor-free Dyck words, arXiv:1606.02183 [math.CO], 2016.
Daniel Birmajer, Juan B. Gil and Michael D. Weiner, On rational Dyck paths and the enumeration of factor-free Dyck words, Discrete Applied Mathematics, 244 (2018), 36-43.
P. Duchon, On the enumeration and generation of generalized Dyck words, Discrete Mathematics, 225 (2000), 121-135.

Cf. A005807 (slope 3/2), A274052 (slope 5/2), A274244 (slope 7/2), A274256 (slope 9/2), A274258 (slope 5/3), A274259 (slope 7/3).

m = 20; f[_] = 0;
Do[f[x_] = (1/(x+1)^4)(-(x^2 (x+1) f[x]^4) + x f[x]^6 + (x-1) x f[x]^5 - (x - 3) x (x+1)^2 f[x]^3 + x (x+1)^3 f[x]^2 + (x+1)^5) + O[x]^m, {m}];
CoefficientList[f[x], x] (* Jean-François Alcover, Sep 28 2019 *)

G.f. satisfies: 0 = x*f^6 + x*(x-1)*f^5 - x^2*(x+1)*f^4 - x*(x-3)*(x+1)^2*f^3 + x*(x+1)^3*f^2 - (x+1)^4*f + (x+1)^5. - Michael D. Weiner, Jan 14 2019

Conjectural o.g.f.: Let E(x) = exp( Sum_{n >= 1} binomial(7*n, 3*n)*x^n/n ). Then A(x) = ( x/series reversion of x*E(x) )^(1/7) = 1 + 5*x + 52*x^2 + 880*x^3 + .... Equivalently, [x^n]( A(x)^(7*n) ) = binomial(7*n, 3*n) for n = 0,1,2,.... - Peter Bala, Jan 01 2020

A274259 Number of factor-free Dyck words with slope 7/3 and length 10n.

Original entry on oeis.org

1, 12, 570, 44689, 4223479, 441010458, 49014411306, 5685822210429, 680500195656621, 83406972284096638, 10416465145620729162, 1320749077779826216029, 169570747575202480367168, 22000830732097549119672094, 2880094468241888675318895339, 379941591968957300338548388051, 50458777676743899501139029335858

Offset: 0

Michael D. Weiner, Jun 16 2016

nonn

a(n) is the number of lattice paths (allowing only north and east steps) starting at (0,0) and ending at (3n,7n) that stay below the line y=7/3x and also do not contain a proper subpath of smaller size.

			a(2) = 570 since there are 570 lattice paths (allowing only north and east steps) starting at (0,0) and ending at (6,14) that stay below the line y=7/3x and also do not contain a proper subpath of small size; e.g., ENNENENNNENNENNNENNN is a factor-free Dyck word but ENNENNENNEENNNNNENNN contains the factor ENNEENNNNN.

Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, On rational Dyck paths and the enumeration of factor-free Dyck words, arXiv:1606.02183 [math.CO], 2016.
P. Duchon, On the enumeration and generation of generalized Dyck words, Discrete Mathematics, 225 (2000), 121-135.

Factor-free Dyck words: A005807 (slope 3/2), A274052 (slope 5/2), A274244 (slope 7/2), A274256 (slope 9/2), A274257 (slope 4/3), A274258 (slope 5/3).

Conjectural o.g.f.: Let E(x) = exp( Sum_{n >= 1} binomial(10*n, 3*n)*x^n/n ). Then A(x) = ( x/series reversion of x*E(x) )^(1/10) = 1 + 12*x + 570*x^2 + 44689*x^3 + ... . Equivalently, [x^n]( A(x)^(10*n) ) = binomial(10*n, 3*n) for n = 0,1,2,... . - Peter Bala, Jan 03 2020

A300388 The number of paths of length 11n from the origin to the line y = 2x/9 with unit East and North steps that stay below the line or touch it.

Original entry on oeis.org

1, 5, 345, 35246, 4255288, 563796161, 79264265868, 11612106079203, 1753402118587333, 270965910076404428, 42648418241303137766, 6813002989827352100145, 1101807202785456951146158, 180034116076502209139781574, 29677341363243548521326632028, 4929368173228370040701922315332

Offset: 0

Bryan T. Ek, Mar 04 2018

Equivalent to nonnegative walks from (0,0) to (11*n,0) with step set [1,2], [1,-9].

			For n=1, the walks are EEEEEEEEENN, EEEEEEEENEN, EEEEEEENEEN, EEEEEENEEEN, EEEEENEEEEN.

M. T. L. Bizley, Derivation of a new formula for the number of minimal lattice paths from (0, 0) to (km, kn) having just t contacts with the line my = nx and having no points above this line; and a proof of Grossman's formula for the number of paths which may touch but do not rise above this line, Journal of the Institute of Actuaries, Vol. 80, No. 1 (1954): 55-62. [Cached copy]
Bryan Ek, Lattice Walk Enumeration, arXiv:1803.10920 [math.CO], 2018.
Bryan Ek, Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics, arXiv:1804.05933 [math.CO], 2018.

Cf. A001764, A060941, A300386, A300387, A300389, A274256.

terms = 16; f[_] = 0;
Do[f[t_] = f[t]^55 t^5 + 4 f[t]^46 t^4 - f[t]^45 t^4 + 5 f[t]^44 t^4 + 6 f[t]^37 t^3 - 3 f[t]^36 t^3 + 12 f[t]^35 t^3 - 4 f[t]^34 t^3 + 10 f[t]^33 t^3 + 4 f[t]^28 t^2 - 3 f[t]^27 t^2 + 9 f[t]^26 t^2 - 6 f[t]^25 t^2 + 12 f[t]^24 t^2 - 6 f[t]^23 t^2 + 10 f[t]^22 t^2 + f[t]^19 t - f[t]^18 t + 2 f[t]^17 t - 2 f[t]^16 t + 3 f[t]^15 t - 3 f[t]^14 t + 4 f[t]^13 t - 4 f[t]^12 t + 5 f[t]^11 t + 1 + O[t]^terms, {terms}];
CoefficientList[f[t], t] (* Jean-François Alcover, Dec 04 2018 *)
nmax = 20; CoefficientList[Series[Exp[Sum[Binomial[11*k, 2*k]*x^k/(11*k), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 16 2021 *)

G.f. satisfies: f = f^55*t^5 + 4*f^46*t^4 - f^45*t^4 + 5*f^44*t^4 + 6*f^37*t^3 - 3*f^36*t^3 + 12*f^35*t^3 - 4*f^34*t^3 + 10*f^33*t^3 + 4*f^28*t^2 - 3*f^27*t^2 + 9*f^26*t^2 - 6*f^25*t^2 + 12*f^24*t^2 - 6*f^23*t^2 + 10*f^22*t^2 + f^19*t - f^18*t + 2*f^17*t - 2*f^16*t + 3*f^15*t - 3*f^14*t + 4*f^13*t - 4*f^12*t + 5*f^11*t + 1.
From Peter Bala, Jan 03 2019: (Start)
O.g.f.: A(x) = exp( Sum_{n >= 1} (1/11)*binomial(11*n, 2*n)*x^n/n ) - Bizley. Cf. A274256.
Recurrence: a(0) = 1 and a(n) = (1/n) * Sum_{k = 0..n-1} (1/11)*binomial(11*n-11*k, 2*n-2*k)*a(k) for n >= 1. (End)
The sequence defined by b(n) := [x^n] A(x)^n begins [1, 5, 715, 116213, 19954187, 3532860880, 637870220023, 116749388814357, ...] and conjecturally satisfies the congruence b(p) == b(1) (mod p^3) for prime p >= 5 except for p = 11 (checked up to p = 101). - Peter Bala, Sep 14 2021
a(n) ~ c * 11^(11*n) / (n^(3/2) * 2^(2*n) * 3^(18*n)), where c = 0.0304820662333129164912550234496338371466905844787974500412037592866845093... - Vaclav Kotesovec, Sep 16 2021

A274258 Number of factor-free Dyck words with slope 5/3 and length 8n.

Original entry on oeis.org

1, 7, 133, 4140, 154938, 6398717, 281086555, 12882897819, 609038885805, 29481041746958, 1453894927584477, 72789271870852237, 3689808842747726368, 189006099916444293090, 9768094831949586349262, 508712466332195692590121, 26670630123516854616641671, 1406503552584980596900001922, 74559627811441047591493767590

Offset: 0

Michael D. Weiner, Jun 16 2016

nonn

a(n) is the number of lattice paths (allowing only north and east steps) starting at (0,0) and ending at (3n,5n) that stay below the line y=5/3x and also do not contain a proper subpath of smaller size.

			a(2) = 133 since there are 133 lattice paths (allowing only north and east steps) starting at (0,0) and ending at (6,10) that stay below the line y=5/3x and also do not contain a proper subpath of small size; e.g., ENEEEENNNNENNNNN is a factor-free Dyck word but ENEENNENNNEENNNN contains the factor EENNENNN.

Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, On rational Dyck paths and the enumeration of factor-free Dyck words, arXiv:1606.02183 [math.CO], 2016.
P. Duchon, On the enumeration and generation of generalized Dyck words, Discrete Mathematics, 225 (2000), 121-135.

Factor-free Dyck words: A005807 (slope 3/2), A274052 (slope 5/2), A274244 (slope 7/2), A274256 (slope 9/2), A274257 (slope 4/3), A274259 (slope 7/3).

Conjectural o.g.f.: Let E(x) = exp( Sum_{n >= 1} binomial(8*n, 3*n)*x^n/n ). Then A(x) = ( x/series reversion of x*E(x) )^(1/8) = 1 + 7*x + 133*x^2 + 4140*x^3 + ... . Equivalently, [x^n]( A(x)^(8*n) ) = binomial(8*n, 3*n) for n = 0,1,2,... . - Peter Bala, Jan 01 2020

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A274052 Number of factor-free Dyck words with slope 5/2 and length 7n.

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A274244 Number of factor-free Dyck words with slope 7/2 and length 9n.

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A274257 Number of factor-free Dyck words with slope 4/3 and length 7n.

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A274259 Number of factor-free Dyck words with slope 7/3 and length 10n.

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A300388 The number of paths of length 11*n from the origin to the line y = 2*x/9 with unit East and North steps that stay below the line or touch it.

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A274258 Number of factor-free Dyck words with slope 5/3 and length 8n.

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A300388 The number of paths of length 11n from the origin to the line y = 2x/9 with unit East and North steps that stay below the line or touch it.