A293946 -id:A293946 - cp's OEIS frontend

A060941 Duchon's numbers: the number of paths of length 5n from the origin to the line y = 2x/3 with unit East and North steps that stay below the line or touch it.

1, 2, 23, 377, 7229, 151491, 3361598, 77635093, 1846620581, 44930294909, 1113015378438, 27976770344941, 711771461238122, 18293652115906958, 474274581883631615, 12388371266483017545, 325714829431573496525, 8613086428709348334675, 228925936056388155632081

Offset: 0

Philippe Flajolet, May 12 2001

A generalization of the ballot numbers.

Alois P. Heinz, Table of n, a(n) for n = 0..500
C. Banderier, Home page
Cyril Banderier and Philippe Flajolet, Basic Analytic Combinatorics of Lattice Paths, Theoret. Comput. Sci. 281 (2002), 37-80.
D. Bevan, D. Levin, P. Nugent, J. Pantone, and L. Pudwell, Pattern avoidance in forests of binary shrubs, arXiv preprint arXiv:1510:08036 [math.CO], 2015.
Daniel Birmajer, Juan B. Gil, Peter R. W. McNamara, and Michael D. Weiner, Enumeration of colored Dyck paths via partial Bell polynomials, arXiv:1602.03550 [math.CO], 2016.
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.
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]
M. T. L. Bizley, Annotated copy of page 59
M. Bousquet-Mélou and A. Jehanne, Polynomial equations with one catalytic variable, algebraic series and map enumeration, arXiv:math/0504018 [math.CO], 2005.
P. Duchon, Home Page
Philippe Duchon, On the enumeration and generation of generalized Dyck words, Discrete Mathematics 225, 2000, 121-135.
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.
P. Flajolet, Home page
Don Knuth, 20th Anniversary Christmas Tree Lecture [A060941 is mentioned after about 65 minutes - _N. J. A. Sloane_, Dec 09 2014]
Michael Wallner, Combinatorics of lattice paths and tree-like structures (Dissertation, Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien), 2016.

Cf. A000108, A001450, A001764, A002294, A300386 - A300389, A322634.

See A293946 for a closely related sequence, also from the Bizley paper.

[&+[1/(5*n+i+1)*Binomial(5*n+1, n-i)*Binomial(5*n+2*i, i): i in [0..n]]: n in [0..30]]; // Vincenzo Librandi, Feb 12 2016

A060941 := n -> hypergeom([-n,5*n/2+1/2,5*n/2+1],[4*n+2,5*n+2],-4)* binomial(5*n,n)/(4*n+1); seq(simplify(A060941(n)),n=0..18); # Peter Luschny, Oct 05 2014

a[n_] := ((5n)!*(5n + 1)!*HypergeometricPFQRegularized[{-n, 5n/2 + 1/2, 5n/2 + 1}, {4n + 2, 5n + 2}, -4])/n!; a /@ Range[0, 16]
(* Jean-François Alcover, Jun 30 2011, after given formula *)

A060941 = lambda n : hypergeometric([-n,5*n/2+1/2,5*n/2+1],[4*n+2,5*n+2],-4)*gamma(1+5*n)/(gamma(1+n)*gamma(2+4*n))
[A060941(n).simplify() for n in range(19)] # Peter Luschny, Oct 05 2014

a(n) = Sum_{i=0..n} 1/(5*n+i+1) * C(5*n+1, n-i) * C(5*n+2*i, i).
a(n) = Sum_{i=0..2*n} (-1)^i/(5*i+1) * C((5*i+1)/2, i) * 1/(1+5*(2*n-i)) * C((1+5*(2*n-i))/2, 2*n-i).
G.f. A(z) satisfies: A(z) = 1+2*z*A^5-z*A^6+z*A^7+z^2*A^10. [Corrected by Bryan T. Ek, Oct 30 2017]
G.f.: A(z) = exp(C(5,2)*z/5 + C(10,4)*z^2/10 + C(15,6)*z^3/15 + ...). - Don Knuth, Oct 05 2014
Recurrence: 216*(n-1)*n*(2*n-1)*(3*n-4)*(3*n-2)*(3*n-1)*(3*n+1)*(6*n-1)*(6*n+1)*(5625*n^4 - 38550*n^3 + 97425*n^2 - 107784*n + 44044)*a(n) = 540*(n-1)*(3*n-4)*(3*n-2)*(126562500*n^10 - 1373625000*n^9 + 6557484375*n^8 - 18192221250*n^7 + 32549973750*n^6 - 39248008800*n^5 + 32203028675*n^4 - 17641491134*n^3 + 6113558828*n^2 - 1191132600*n + 96112128)*a(n-1) - 450*(5*n-9)*(5*n-8)*(5*n-7)*(5*n-6)*(63281250*n^9 - 718453125*n^8 + 3556125000*n^7 - 10046426250*n^6 + 17765816250*n^5 - 20240090325*n^4 + 14698993900*n^3 - 6468702396*n^2 + 1533535184*n - 142988160)*a(n-2) + 78125*(n-2)*(5*n-14)*(5*n-13)*(5*n-12)*(5*n-11)*(5*n-9)*(5*n-8)*(5*n-7)*(5*n-6)*(5625*n^4 - 16050*n^3 + 15525*n^2 - 6084*n + 760)*a(n-3). - Vaclav Kotesovec, Oct 05 2014
Asymptotics (Duchon, 2000): a(n) ~ c * (3125/108)^n / n^(3/2), where c = 0.0876612192439026461763141944768209255550234422281635788... (constant corrected, in the reference "On the enumeration and generation of generalized Dyck words", p.132 is a wrong value 0.0887). - Vaclav Kotesovec, Oct 05 2014, c = sqrt(5*(10^(2/3) - 5^(1/3)/2^(2/3) - 2))/(18*sqrt(Pi)). - Vaclav Kotesovec, Sep 16 2021
a(n) = Gamma(n+4/5)*Gamma(n+3/5)*Gamma(n+2/5)*3125^n*hypergeom([-n, (5/2)*n+1, (5/2)*n+1/2], [5*n+2, 4*n+2], -4)*Gamma(n+1/5)/ (Pi^2*csc((2/5)*Pi)*csc((1/5)*Pi)*Gamma(4*n+2)). - Robert Israel, Oct 05 2014
a(n) = A002294(n)*hypergeom([-n,5*n/2+1/2,5*n/2+1],[4*n+2,5*n+2],-4). - Peter Luschny, Oct 05 2014
O.g.f. A(x) satisfies: A(x)^5 = 1/x*series reversion( x/((1+x)*C(x))^5 ), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. See A001450. - Peter Bala, Oct 05 2015
The sequence defined by b(n) := [x^n] A(x)^n begins [1, 2, 50, 1415, 42258, 1300727, 40820837, 1298493730, ...] and conjecturally satisfies the congruence b(p) == b(1) (mod p^3) for prime p >= 7 (checked up to p = 101). [Added 23 Oct 2024: More generally, let r  be an integer and s a positive integer and define a sequence u(n) by u(n) = [x^(s*n)] A(x)^(r*n). Then we conjecture that the supercongruences u(n*p^k) == u(n*p^(k-1)) (mod p^(3*k)) hold for all primes p >= 7 and positive integers n and k.] - Peter Bala, Sep 12 2021
Inductively define a family of sequences {a(i,n) : n >= 0}, i >= 1, by setting a(1,n) = a(n) and, for i >= 2, a(i,n) = [x^n] ( exp(Sum_{k >= 1} a(i-1,k)*x^k/k) )^n. We conjecture that the sequences {a(i,n) : n >= 0}, i >= 2, also satisfy the supercongruences u(n*p^k) == u(n*p^(k-1)) (mod p^(3*k)) for primes p >= 7 and positive integers n and k. - Peter Bala, Oct 24 2024

A100982 Number of admissible sequences of order j; related to 3x+1 problem and Wagon's constant.

Original entry on oeis.org

1, 1, 2, 3, 7, 12, 30, 85, 173, 476, 961, 2652, 8045, 17637, 51033, 108950, 312455, 663535, 1900470, 5936673, 13472296, 39993895, 87986917, 257978502, 820236724, 1899474678, 5723030586, 12809477536, 38036848410, 84141805077, 248369601964

Offset: 1

Steven Finch, Jan 13 2005

Eric Roosendaal counted all admissible sequences up to order j=1000 (2005). Note: there is a typo in both Wagon and Chamberland in the definition of Wagon's constant 9.477955... The expression floor(1 + 2*i + i*log_2(3)) should be replaced by floor(1 + i + i*log_2(3)).
The length of all admissible sequences of order j is A020914(j). - T. D. Noe, Sep 11 2006
Conjecture: a(n) is given for each n > 3 by a formula using a(2)..a(n-1). This allows us to create an iterative algorithm which generates a(n) for each n > 6. This has been proved for each n <= 53. For higher values of n the algorithm must be slightly modified. - Mike Winkler, Jan 03 2018
Theorem 1: a(k) is given for each k > 1 by a formula using a(1)..a(k-1). Namely, a(1)=1 and a(k+1) = Sum_{m=1..k} (-1)^(m-1)*binomial(floor((k-m+1)*(log(3)/log(2))) + m - 1, m)*a(k-m+1) for k >= 1. - Vladimir M. Zarubin, Sep 25 2015
Theorem 2: a(n) can be generated for each n > 2 algorithmically in a Pascal's triangle-like manner from the two starting values 0 and 1. This result is based on the fact that the Collatz residues (mod 2^k) can be evolved according to a binary tree. There is a direct connection with A076227, A056576 and A022921. - Mike Winkler, Sep 12 2017
A177789 shows another theorem and algorithm for generating a(n). - Mike Winkler, Sep 12 2017

			The unique admissible sequence of order 1 is 3/2, 1/2.
The unique admissible sequence of order 2 is 3/2, 3/2, 1/2, 1/2.
The two admissible sequences of order 3 are 3/2, 3/2, 3/2, 1/2, 1/2 and 3/2, 3/2, 1/2, 3/2, 1/2.
a(13) = 8045 = binomial(floor(5*(13-2)/3), 13-2)
- Sum_{i=2..6} binomial(floor((3*(13-i)+0)/2), 13-i)*a(i)
- Sum_{i=7..11} binomial(floor((3*(13-i)-1)/2), 13-i)*a(i)
- Sum_{i=12..12} binomial(floor((3*(13-i)-2)/2), 13-i)*a(i)
= 31824 - 4368*1 - 3003*2 - 715*3 - 495*7 - 120*12 - 28*30 - 21*85 - 5*173 - 4*476 - 1*961 - 0*2652. (Conjecture)
From _Mike Winkler_, Sep 12 2017: (Start)
The next table shows how Theorem 2 works. No entry is equal to zero.
n =       3  4  5   6   7   8   9  10  11   12 .. |A076227(k)=
--------------------------------------------------|
k =  2 |  1                                       |     1
k =  3 |  1  1                                    |     2
k =  4 |     2  1                                 |     3
k =  5 |        3   1                             |     4
k =  6 |        3   4   1                         |     8
k =  7 |            7   5   1                     |    13
k =  8 |               12   6   1                 |    19
k =  9 |               12  18   7   1             |    38
k = 10 |                   30  25   8   1         |    64
k = 11 |                   30  55  33   9    1    |   128
:      |                        :   :   :    : .. |    :
--------------------------------------------------|---------
a(n) =    2  3  7  12  30  85 173 476 961 2652 .. |
The entries (k,n) in this table are generated by the rule (k+1,n) = (k,n) + (k,n-1). The last value of (k+1,n) is given by k+1 = A056576(n-1), or the highest value in column n is given twice only if A022921(n-2) = 2. Then a(n) is equal to the sum of the entries in column n. For n = 7 there is 1 = 0 + 1, 5 = 1 + 4, 12 = 5 + 7, 12 = 12 + 0. Therefore a(7) = 1 + 5 + 12 + 12 = 30. The sum of row k is equal to A076227(k). (End)
From _Ruud H.G. van Tol_, Dec 04 2023: (Start)
A tree view.
n-tree--A098294--ids-----paths-----------------a(n)
0 ._          0  0       0                       -
1 |_          1  1       10                      1
2 |_._        2  2       1100                    1
3 |_|_        2  3-4     11010     -   11100     2
4 |_|_._      3  5-7     1101100   -  1111000    3
5 |_|_|_      3  8-14    11011010  - 11111000    7
6 |_|_|_._    4  15-26   1101101100-1111110000  12
7 |_|_|_|_._  5  27-56   ...                    30
8 |_|_|_|_|_  5  57-141  ...                    85
...
For n>=1, the endpoints are at A098294(n) to the right.
(End)

T. D. Noe, Table of n, a(n) for n = 1..500
M. Chamberland, Una actualizacio del problema 3x+1, Butl. Soc. Catalana Mat. 18 (2003) 19-45.
M. Chamberland, English translation
Ruud H.G. van Tol, Sequence as counts of lattice walks
Ruud H.G. van Tol, A100982 Musings
Stan Wagon, The Collatz problem, Math. Intelligencer 7 (1985) 72-76.
Mike Winkler, On a stopping time algorithm of the 3n + 1 function, 2011.
Mike Winkler, On the structure and the behaviour of Collatz 3n + 1 sequences - Finite subsequences and the role of the Fibonacci sequence, arXiv:1412.0519 [math.GM], 2014.
Mike Winkler, New results on the stopping time behaviour of the Collatz 3x + 1 function, arXiv:1504.00212 [math.GM], 2015.
Mike Winkler, The algorithmic structure of the finite stopping time behavior of the 3x + 1 function, arXiv:1709.03385 [math.GM], 2017.

Cf. A122790 (Wagon's constant), A076227, A056576, A022921, A098294, A177789.

Cf. A060941, A293946.

(* based on Eric Roosendaal's algorithm *) nn=100; Clear[x,y]; Do[x[i]=0, {i,0,nn+1}]; x[1]=1; t=Table[Do[y[cnt]=x[cnt]+x[cnt-1], {cnt,p+1}]; Do[x[cnt]=y[cnt], {cnt,p+1}]; admis=0; Do[If[(p+1-cnt)*Log[3]T. D. Noe, Sep 11 2006 *)

/* translation of the above code from T. D. Noe */
{limit=100; n=1; x=y=vector(limit+1); x[1]=1; for(b=2, limit, for(c=2, b+1, y[c]=x[c]+x[c-1]); for(c=2, b+1, x[c]=y[c]); a_n=0; for(c=1, b+1, if((b+1-c)*log(3)Mike Winkler, Feb 28 2015

/* algorithm for the Conjecture */
{limit=53; zn=vector(limit); zn[2]=1; zn[3]=2; zn[4]=3; zn[5]=7; zn[6]=12; f=1; e1=-1; e2=-2; for(n=7, limit, m=floor((n-1)*log(3)/log(2))-(n-1); j=(m+n-2)!/(m!*(n-2)!); if(n>6*f, if(frac(n/2)==0, e=e1, e=e2)); if(frac((n-6 )/12)==0, f++; e1=e1+2); if(frac((n-12)/12)==0, f++; e2=e2+2); Sum=a=b=0; c=1; d=5; until(c>=n-1, for(i=2+a*5+b, 1+d+a*5, if(i>11 && frac((i+2)/6)==0, b++); delta=e-a; Sum=Sum+binomial(floor((3*(n-i)+delta)/2),n-i)*zn[i]; c++); a++; for(k=3, 50, if(n>=k*6 && a==k-1, d=k+3))); zn[n]=j-Sum; print(n" "zn[n]))} \\ Mike Winkler, Jan 03 2018

/* cf. code for Theorem 2 */
{limit=100; /*or limit>100*/ p=q=vector(limit); c=2; w=log(3)/log(2); for(n=3, limit, p[1]=Sum=1; for(i=2, c, p[i]=p[i-1]+q[i]; Sum=Sum+p[i]); a_n=Sum; print(n" "a_n); for(i=1, c, q[i]=p[i]); d=floor(n*w)-floor((n-1)*w); if(d==2, c++)); } \\ Mike Winkler, Apr 14 2015

/* algorithm for Theorem 1 */
n=20; a=vector(n); log32=log(3)/log(2);
{a[1]=1; for ( k=1, n-1, a[k+1]=sum( m=1,k,(-1)^(m-1)*binomial( floor( (k-m+1)*log32)+m-1,m)*a[k-m+1] ); print(k" "a[k]) );
} \\ Vladimir M. Zarubin, Sep 25 2015

/* algorithm for Theorem 2 */
{limit=30; /*or limit>30*/ R=matrix(limit,limit); R[2,1]=0; R[2,2]=1; for(n=2, limit, print; print1("For n="n" in column n: "); Kappa_n=floor(n*log(3)/log(2)); a_n=0; for(k=n, Kappa_n, R[k+1,n]=R[k,n]+R[k,n-1]; print1(R[k+1,n]", "); a_n=a_n+R[k+1,n]); print; print(" and the sum is a(n)="a_n))} \\ Mike Winkler, Sep 12 2017

A sequence s(k), where k=1, 2, ..., n, is *admissible* if it satisfies s(k)=3/2 exactly j times, s(k)=1/2 exactly n-j times, s(1)*s(2)*...*s(n) < 1 but s(1)*s(2)*...*s(m) > 1 for all 1 < m < n.
a(n) = (m+n-2)!/(m!*(n-2)!) - Sum_{i=2..n-1} binomial(floor((3*(n-i)+b)/2), n-i)*a(i), where m = floor((n-1)*log_2(3))-(n-1) and b assumes different integer values within the sum at intervals of 5 or 6 terms. (Conjecture)
a(n) = Sum_{k=n-1..A056576(n-1)} (k,n). (Theorem 2, cf. example)
a(k) = 2*A076227(A020914(k)-1) - A076227(A020914(k)), for k > 0. - Vladimir M. Zarubin, Sep 29 2019
a(1)=1, a(n) = Sum_{k=0..A020914(n-1)-n-2} A325904(k)*binomial(A020914(n-1)-k-2, n-2) for n>1. - Benjamin Lombardo, Oct 18 2019

Two more terms from Jules Renucci (jules.renucci(AT)wanadoo.fr), Nov 02 2005

More terms from T. D. Noe, Sep 11 2006

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

A322634 Sum of attendance numbers of all histories of length 5*n in the Bizley-Duchon's club model, divided by 5.

Original entry on oeis.org

5, 153, 4537, 133189, 3891675, 113415423, 3299905647

Offset: 1

Hugo Pfoertner, Dec 22 2018

The Bizley-Duchon's club model is equivalent to the lattice paths from (0,0) to (3*n,2*n) described in A293946. The attendance history of the club consists of persons entering in pairs and leaving in groups of three. The club closes when no persons are remaining. a(k)/A293946(k) is proportional to the mean area under the "filling level curve" of the club.

Banderier et al. show that the mean area is asymptotic to K*n^(3/2), with K=(1/2)*(15*Pi)^(1/2).

			a(1) = (15 + 10)/5 = 5:
  Contributions of the A293946(1) = 2 attendance histories are
  0 (+2) 2 (+2) 4 (+2) 6 (-3) 3 (-3) 0 -> 2 + 4 + 6 + 3 = 15
  0 (+2) 2 (+2) 4 (-3) 1 (+2) 3 (-3) 0 -> 2 + 4 + 1 + 3 = 10.

Cyril Banderier, Bernhard Gittenberger, Analytic Combinatorics of Lattice Paths: Enumeration and Asymptotics for the Area. Chassaing, Philippe and others. Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, 2006, Nancy, France. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AG, Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, pp.345-356, 2006, DMTCS Proceedings.
Cyril Banderier, Michael Wallner, Lattice paths of slope 2/5, arXiv:1605.02967 [cs.DM], 10 May 2016.

Cf. A060941, A293946.

A294207 Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,k), 0 <= 3k <= 2n, that are below the line 3y=2x, only touching at the end points.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 3, 3, 1, 4, 7, 7, 1, 5, 12, 19, 19, 1, 6, 18, 37, 37, 1, 7, 25, 62, 99, 99, 1, 8, 33, 95, 194, 293, 293, 1, 9, 42, 137, 331, 624, 624, 1, 10, 52, 189, 520, 1144, 1768, 1768, 1, 11, 63, 252, 772, 1916, 3684, 5452, 5452

Offset: 0

Danny Rorabaugh, Oct 24 2017

			The table begins:
n=0: 1;
n=1: 1;
n=2: 1, 1;
n=3: 1, 2,  2;
n=4: 1, 3,  3;
n=5: 1, 4,  7,  7;
n=6: 1, 5, 12, 19,  19;
n=7: 1, 6, 18, 37,  37;
n=8: 1, 7, 25, 62,  99,  99;
n=9: 1, 8, 33, 95, 194, 293, 293.

Eric Weisstein's World of Mathematics, Lattice Path.

Cf. A009766, A293946.

T[, 0] = 1; T[n, k_] := T[n, k] = Which[0 < k < 2(n-1)/3, T[n-1, k] + T[n, k-1], 2(n-1) <= 3k <= 2n, T[n, k-1]];
Table[T[n, k], {n, 0, 15}, {k, 0, Floor[2n/3]}] // Flatten (* Jean-François Alcover, Jul 10 2018 *)

T = [[1]]
for n in range(1,15):
    T.append([T[-1][0]])
    for k in range(1,floor(2*n/3) + 1):
        T[-1].append(T[-1][k-1])
        if 2*(n-1)>3*k:
            T[-1][-1] += T[-2][k]

T(n,0) = 1; for 0 < k < 2(n-1)/3, T(n,k) = T(n-1,k) + T(n,k-1); for 2(n-1) <= 3k <= 2n, T(n,k) = T(n,k-1).

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A060941 Duchon's numbers: the number of paths of length 5*n from the origin to the line y = 2*x/3 with unit East and North steps that stay below the line or touch it.

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A100982 Number of admissible sequences of order j; related to 3x+1 problem and Wagon's constant.

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PARI

PARI

PARI

PARI

PARI

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

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A322634 Sum of attendance numbers of all histories of length 5*n in the Bizley-Duchon's club model, divided by 5.

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A294207 Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,k), 0 <= 3k <= 2n, that are below the line 3y=2x, only touching at the end points.

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A060941 Duchon's numbers: the number of paths of length 5n from the origin to the line y = 2x/3 with unit East and North steps that stay below the line or touch it.