A274244 -id:A274244 - cp's OEIS frontend

A005807 Sum of adjacent Catalan numbers.

2, 3, 7, 19, 56, 174, 561, 1859, 6292, 21658, 75582, 266798, 950912, 3417340, 12369285, 45052515, 165002460, 607283490, 2244901890, 8331383610, 31030387440, 115948830660, 434542177290, 1632963760974, 6151850548776

Offset: 0

N. J. A. Sloane

nonn

The aerated sequence has Hankel transform F(n+2)*F(n+3) (A001654(n+2)). - Paul Barry, Nov 04 2008

			G.f. = 2 + 3*x+ 7*x^2 + 19*x^3 + 56*x^4 + 174*x^5 + 561*x^6 + 1859*x^7 + ...

D. E. Knuth, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Daniel Birmajer, Juan B. Gil, Michael D. Weiner, On rational Dyck paths and the enumeration of factor-free Dyck words, arXiv:1606.02183 [math.CO], (7-June-2016); see p. 9
Aleksandar Cvetkovic, Predrag Rajkovic and Milos Ivkovic, Catalan Numbers, the Hankel Transform and Fibonacci Numbers, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.3
Sergio Falcon, Catalan transform of the K-Fibonacci sequence, Commun. Korean Math. Soc. 28 (2013), No. 4, pp. 827-832; http://dx.doi.org/10.4134/CKMS.2013.28.4.827.
Manuel Flores, Yuta Kimura, Baptiste Rognerud, Combinatorics of quasi-hereditary structures, arXiv:2004.04726 [math.RT], 2020.
Guo-Niu Han, Enumeration of Standard Puzzles
Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 431
Qi Wang, Tau-tilting finite simply connected algebras, arXiv:1910.01937 [math.RT], 2019.
Fumitaka Yura, Hankel Determinant Solution for Elliptic Sequence, arXiv:1411.6972 [nlin.SI], (25-November-2014); see p. 7.

Cf. A000108.

Cf. A071716, A000778, A060941, A274052, A274244.

[((5*n+4)*Factorial(2*n))/(Factorial(n)*Factorial(n+2)): n in [0..30] ];  // Vincenzo Librandi, Aug 19 2011

A005807List := proc(m) local A, P, n; A := [2,3]; P := [2,3];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-2]]);
A := [op(A), P[-1]] od; A end: A005807List(25); # Peter Luschny, Mar 26 2022

a[n_]:=Binomial[2*n, n]*(5*n+4)/(n+1)/(n+2); (* Vladimir Joseph Stephan Orlovsky, Dec 13 2008 *)
a[ n_] := If[ n < 0, 0, CatalanNumber[n] + CatalanNumber[n + 1]]; (* Michael Somos, Jan 17 2015 *)
Total/@Partition[CatalanNumber[Range[0,30]],2,1] (* Harvey P. Dale, Jun 21 2025 *)

{a(n) = if( n<0, 0, binomial(2*n, n) * (5*n+4) / ((n+1) * (n+2)))};

from _future_ import division
A005807_list, b = [], 2
for n in range(10**3):
    A005807_list.append(b)
    b = b*(4*n+2)*(5*n+9)//((n+3)*(5*n+4)) # Chai Wah Wu, Jan 28 2016

[catalan_number(i)+catalan_number(i+1) for i in range(0,25)] # Zerinvary Lajos, May 17 2009

a(n) = C(n)+C(n+1) = ((5*n+4)*(2*n)!)/(n!*(n+2)!).
G.f. A(x) satisfies x^2*A(x)^2 + (x-1)*A(x) + (x+2) = 0. - Michael Somos, Sep 11 2003
G.f.: (1-x - (1+x)*sqrt(1-4*x)) / (2*x^2) = (4+2*x) / (1-x + (1+x)*sqrt(1-4*x)). a(n)*(n+2)*(5*n-1) = a(n-1)*2*(2*n-1)*(5*n+4), n>0. - Michael Somos, Sep 11 2003
a(n) ~ 5*Pi^(-1/2)*n^(-3/2)*2^(2*n)*{1 - 93/40*n^-1 + 625/128*n^-2 - 10227/1024*n^-3 + 661899/32768*n^-4 ...}. - Joe Keane (jgk(AT)jgk.org), Sep 13 2002
G.f.: c(x)*(1+c(x))= (-1 +(1+x)*c(x))/x with the g.f. c(x) of A000108 (Catalan).
a(n) = binomial(2*n,n)/(n+1)*hypergeom([-1,n+1/2],[n+2],-4). - Peter Luschny, Aug 15 2012
D-finite with recurrence (n+2)*a(n) + (-3*n-2)*a(n-1) + 2*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Dec 02 2012
0 = a(n)*(+16*a(n+1) + 38*a(n+2) - 18*a(n+3)) + a(n+1)*(-14*a(n+1) + 19*a(n+2) - 7*a(n+3)) + a(n+2)*(+a(n+2) + a(n+3)) for all n>=0. - Michael Somos, Jan 17 2015
0 = a(n)^2*(+368*a(n+1) - 182*a(n+2)) + a(n)*a(n+1)*(-306*a(n+1) + 317*a(n+2)) + a(n)*a(n+2)*(-77*a(n+2)) + a(n+1)^2*(-14*a(n+1) - 6*a(n+2)) + a(n+1)*a(n+2)*(+8*a(n+2)) for all n>=0. - Michael Somos, Jan 17 2015
E.g.f.: (BesselI(0,2*x) - (x - 1)*BesselI(1,2*x)/x)*exp(2*x). - Ilya Gutkovskiy, Jun 08 2016
G.f. with 1 prepended: Let E(x) = exp( Sum_{n >= 1} binomial(5*n,2*n)*x^n/n ). Then A(x) = ( x/series reversion of x*E(x) )^(1/5) = ( x/series reversion of x*D(x)^5 )^(1/5), where D(x) = 1 + 2*x + 23*x^2 + 371*x^3 + ... is the o.g.f. for A060941 .... Cf. A274052 and A274244. - Peter Bala, Jan 01 2020

More terms from Joe Keane (jgk(AT)jgk.org), Feb 08 2000

Asymptotic series corrected and extended by Michael Somos, Sep 11 2003

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

Original entry on oeis.org

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

A274256 Number of factor-free Dyck words with slope 9/2 and length 11n.

Original entry on oeis.org

1, 5, 70, 1696, 49493, 1593861, 54591225, 1950653202, 71889214644, 2712628146949, 104277713515456, 4069334248174800, 160785480249706192, 6419443865094494044, 258585021917711797850, 10496205397574996367474, 428899108081734423242550, 17628723180468295514015268, 728347675604866545590505024

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 (2n,9n) that stay below the line y=9/2x and also do not contain a proper subpath of smaller size.

			a(2) = 70 since there are 70 lattice paths (allowing only north and east steps) starting at (0,0) and ending at (4,18) that stay below the line y=9/2x and also do not contain a proper subpath of small size; e.g., ENNENNENNNNNNENNNNNNNN is a factor-free Dyck word but ENEENENNNNNNNNNNNNNNNN contains the factor ENENNNNNNNN.

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), A274257 (slope 4/3), A274258 (slope 5/3), A274259 (slope 7/3).

Conjectural o.g.f.: Let E(x) = exp( Sum_{n >= 1} binomial(11*n,2*n)*x^n/n ). Then A(x) = ( x/series reversion of x*E(x) )^(1/11) = 1 + 5*x + 70*x^2 + 1696*x^3 + ... . Equivalently, [x^n]( A(x)^(11*n) ) = binomial(11*n, 2*n) for n = 0,1,2,... . - Peter Bala, Jan 01 2020

A300387 The number of paths of length 9n from the origin to the line y = 2x/7 with unit East and North steps that stay below the line or touch it.

Original entry on oeis.org

1, 4, 178, 11654, 900239, 76266406, 6853777795, 641688752961, 61916364799849, 6113859987916630, 614832988424140624, 62752222758863566993, 6483650829899569496380, 676834416167597357806799, 71278487569046416052210050, 7563527671079260544924794587, 807900192360879042402313084390

Offset: 0

Bryan T. Ek, Mar 04 2018

Equivalent to nonnegative walks from (0,0) to (9*n,0) with step set [1,2], [1,-7].

			For n=1, the possible walks are EEEEEEENN, EEEEEENEN, EEEEENEEN, EEEENEEEN.

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, A300388, A300389, A274244.

terms = 17; f[_] = 0;
Do[f[t_] = f[t]^36 t^4 + 3 f[t]^29 t^3 - f[t]^28 t^3 + 4 f[t]^27 t^3 + 3 f[t]^22 t^2 - 2 f[t]^21 t^2 + 6 f[t]^20 t^2 - 3 f[t]^19 t^2 + 6 f[t]^18 t^2 + f[t]^15 t - f[t]^14 t + 2 f[t]^13 t - 2 f[t]^12 t + 3 f[t]^11 t - 3 f[t]^10 t + 4 f[t]^9 t + 1 + O[t]^terms, {terms}];
CoefficientList[f[t], t] (* Jean-François Alcover, Dec 04 2018 *)
nmax = 20; CoefficientList[Series[Exp[Sum[Binomial[9*k, 2*k]*x^k/(9*k), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 16 2021 *)

G.f. satisfies: f=f^36*t^4+3*f^29*t^3-f^28*t^3+4*f^27*t^3+3*f^22*t^2-2*f^21*t^2+6*f^20*t^2-3*f^19*t^2+6*f^18*t^2+f^15*t-f^14*t+2*f^13*t-2*f^12*t+3*f^11*t-3*f^10*t+4*f^9*t+1.
From Peter Bala, Jan 03 2019: (Start)
O.g.f.: A(x) = exp( Sum_{n >= 1} (1/9)*binomial(9*n, 2*n)*x^n/n ) - Bizley. Cf. A274244.
Recurrence: a(0) = 1 and a(n) = (1/n) * Sum_{k = 0..n-1} (1/9)*binomial(9*n-9*k, 2*n-2*k)*a(k) for n >= 1. (End)
The sequence defined by b(n) := [x^n] A(x)^n begins [1, 4, 372, 39298, 4384884, 504464254, 59183637186, 7038517648243, ...] and conjecturally satisfies the congruence b(p) == b(1) (mod p^3) for prime p >= 11 (checked up to p = 101). - Peter Bala, Sep 14 2021
a(n) ~ c * 3^(18*n) / (n^(3/2) * 2^(2*n) * 7^(7*n)), where c = 0.0389180896257538883301359279112039841187646397413254619045749515282872957... - Vaclav Kotesovec, Sep 16 2021

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

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

cp's OEIS Frontend

A005807 Sum of adjacent Catalan numbers.

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

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A274256 Number of factor-free Dyck words with slope 9/2 and length 11n.

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A300387 The number of paths of length 9*n from the origin to the line y = 2*x/7 with unit East and North steps that stay below the line or touch it.

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

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A300387 The number of paths of length 9n from the origin to the line y = 2x/7 with unit East and North steps that stay below the line or touch it.