A322631 -id:A322631 - 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

A322632 Decimal expansion of the real solution to 23x^5 - 41x^4 + 10x^3 - 6x^2 - x - 1 = 0. Constant occurring in the asymptotic behavior of the number of lattice paths of slope 2/5, observed by D. Knuth.

Original entry on oeis.org

1, 6, 3, 0, 2, 5, 7, 6, 6, 2, 9, 9, 0, 3, 5, 0, 1, 4, 0, 4, 2, 4, 8, 0, 1, 8, 4, 9, 3, 1, 5, 9, 8, 6, 3, 0, 0, 5, 1, 4, 5, 8, 4, 4, 2, 6, 6, 9, 0, 1, 4, 9, 4, 0, 5, 8, 4, 9, 8, 5, 0, 2, 6, 5, 9, 5, 2, 5, 6, 8, 9, 1, 2, 9, 8, 6, 8, 5, 0, 4, 7, 9, 8, 3, 4, 1, 3, 2, 4, 1

Offset: 1

Hugo Pfoertner, Dec 21 2018

In his 2014 lecture in Paris "Problems That Philippe (Flajolet) Would Have Loved" D. Knuth discussed as Problem 4 "Lattice Paths of Slope 2/5" and reported as an empirical observation that A[5*t-1,2*t-1]/B[5*t-1,2*t-1] = a - b/t + O(t^-2), with constants a~=1.63026 and b~=0.159. For the meaning of A and B see A322631. The exact values of a (this sequence) and b (A322633) were found in 2016 by Banderier and Wallner.

			1.6302576629903501404248018493159863005145844266901494058498502659525689...

Cyril Banderier, Michael Wallner, Lattice paths of slope 2/5, arXiv:1605.02967 [cs.DM], 10 May 2016.
D. E. Knuth, Problems That Philippe Would Have Loved, Paris 2014.

Cf. A322631, A322633.

evalf[100](solve(23*x^5-41*x^4+10*x^3-6*x^2-x-1=0,x)[1]); # Muniru A Asiru, Dec 21 2018

RealDigits[Root[23#^5 - 41#^4 + 10#^3 - 6#^2 - # - 1&, 1], 10, 100][[1]] (* Jean-François Alcover, Dec 30 2018 *)

solve(x=1,2,23*x^5-41*x^4+10*x^3-6*x^2-x-1)

A322633 Decimal expansion of the real solution to 11571875x^5 - 5363750x^4 + 628250x^3 - 97580x^2 + 5180*x - 142 = 0, multiplied by 3/7. Constant occurring in the asymptotic behavior of the number of lattice paths of slope 2/5, observed by D. Knuth.

Original entry on oeis.org

1, 5, 8, 6, 6, 8, 2, 2, 6, 9, 7, 2, 0, 2, 2, 7, 7, 5, 5, 1, 4, 7, 8, 0, 4, 1, 9, 8, 1, 6, 3, 9, 2, 7, 3, 9, 2, 0, 6, 3, 7, 3, 7, 5, 7, 5, 5, 3, 0, 6, 6, 9, 7, 7, 9, 6, 3, 6, 6, 5, 2, 4, 0, 9, 5, 4, 4, 0, 7, 6, 7, 0, 2, 1, 0, 9, 5, 3, 1, 2, 5, 7, 9, 6, 1, 9, 1, 7, 1, 9, 2

Offset: 0

Hugo Pfoertner, Dec 21 2018

In his 2014 lecture in Paris "Problems That Philippe (Flajolet) Would Have Loved" D. Knuth discussed as Problem 4 "Lattice Paths of Slope 2/5" and reported as an empirical observation that A[5*t-1,2*t-1]/B[5*t-1,2*t-1] = a - b/t + O(t^-2), with constants a~=1.63026 and b~=0.159. For the meaning of A and B see A322631. The exact values of a (A322632) and b (this sequence) were found in 2016 by Banderier and Wallner.

			0.1586682269720227755147804198163927392063737575530669779636652409544...

Cyril Banderier, Michael Wallner, Lattice paths of slope 2/5, arXiv:1605.02967 [cs.DM], 10 May 2016.
Don E. Knuth, Problems That Philippe Would Have Loved, Paris 2014.

Cf. A322631, A322632.

evalf[100]((3/7)*solve(11571875*x^5 -5363750*x^4 +628250*x^3 -97580*x^2 +5180*x -142=0, x)[1]); # Muniru A Asiru, Dec 21 2018

(3/7)*solve(x=0, 1, 11571875*x^5 -5363750*x^4 +628250*x^3 -97580*x^2 +5180*x -142)

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

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A322632 Decimal expansion of the real solution to 23x^5 - 41x^4 + 10x^3 - 6x^2 - x - 1 = 0. Constant occurring in the asymptotic behavior of the number of lattice paths of slope 2/5, observed by D. Knuth.

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A322633 Decimal expansion of the real solution to 11571875x^5 - 5363750x^4 + 628250x^3 - 97580x^2 + 5180*x - 142 = 0, multiplied by 3/7. Constant occurring in the asymptotic behavior of the number of lattice paths of slope 2/5, observed by D. Knuth.

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cp's OEIS Frontend

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

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Author

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Formula

A322632 Decimal expansion of the real solution to 23*x^5 - 41*x^4 + 10*x^3 - 6*x^2 - x - 1 = 0. Constant occurring in the asymptotic behavior of the number of lattice paths of slope 2/5, observed by D. Knuth.

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PARI

A322633 Decimal expansion of the real solution to 11571875*x^5 - 5363750*x^4 + 628250*x^3 - 97580*x^2 + 5180*x - 142 = 0, multiplied by 3/7. Constant occurring in the asymptotic behavior of the number of lattice paths of slope 2/5, observed by D. Knuth.

Views

Author

Keywords

Comments

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Crossrefs

Programs

Maple

PARI

A322632 Decimal expansion of the real solution to 23x^5 - 41x^4 + 10x^3 - 6x^2 - x - 1 = 0. Constant occurring in the asymptotic behavior of the number of lattice paths of slope 2/5, observed by D. Knuth.

A322633 Decimal expansion of the real solution to 11571875x^5 - 5363750x^4 + 628250x^3 - 97580x^2 + 5180*x - 142 = 0, multiplied by 3/7. Constant occurring in the asymptotic behavior of the number of lattice paths of slope 2/5, observed by D. Knuth.