A027307 -id:A027307 - cp's OEIS frontend

A108451 Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and have k pyramids of the first kind (a pyramid of the first kind is a sequence u^pd^p for some positive integer p, starting at the x-axis).

1, 1, 1, 6, 3, 1, 44, 16, 5, 1, 344, 116, 30, 7, 1, 2856, 928, 224, 48, 9, 1, 24816, 7856, 1840, 376, 70, 11, 1, 223016, 69264, 15912, 3184, 580, 96, 13, 1, 2056256, 629472, 142592, 28176, 5080, 844, 126, 15, 1, 19344472, 5855472, 1312360, 256992, 46072

Offset: 0

Emeric Deutsch, Jun 11 2005

Also number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and have k pyramids of the second kind (a pyramid of the second kind is a sequence U^pd^(2p) for some positive integer p, starting at the x-axis). Row sums yield A027307. Column 0 yields A108452. Number of pyramids of the first kind in all paths from (0,0) to (3n,0) is given by A108453.

			T(2,1)=3 because we have (ud)Udd, (uudd) and Udd(ud), the pyramids of the first kind being shown between parentheses.
Triangle begins:
1;
1,1;
6,3,1;
44,16,5,1;

Emeric Deutsch, Problem 10658: Another Type of Lattice Path, American Math. Monthly, 107, 2000, 368-370.

Cf. A027307, A108452, A108453, A108445.

A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3: G:=(1-z)/(1-t*z-z*(1-z)*A*(1+A)): Gser:=simplify(series(G,z=0,13)): P[0]:=1: for n from 1 to 10 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 9 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields sequence in triangular form

G.f.: (1-z)/[1-tz-z(1-z)A(1+A)], where A=1+zA^2+zA^3=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3 (the g.f. of A027307).

A108452 Number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and having no pyramids of the first kind (a pyramid of the first kind is a sequence u^pd^p for some positive integer p, starting at the x-axis).

Original entry on oeis.org

1, 1, 6, 44, 344, 2856, 24816, 223016, 2056256, 19344472, 184956240, 1792088296, 17558218048, 173659691928, 1731556718224, 17387182158184, 175670235597120, 1784561125349464, 18216639085961552, 186762117058304104

Offset: 0

Emeric Deutsch, Jun 11 2005

nonn

Also number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and having no pyramids of the second kind (a pyramid of the second kind is a sequence U^pd^(2p) for some positive integer p, starting at the x-axis). Column 0 of A108451.

			a(2)=6 because the paths uUddd, UddUdd, Ududd, UdUddd, Uuddd and UUdddd have no pyramids of the first kind.

Vaclav Kotesovec, Table of n, a(n) for n = 0..100
Emeric Deutsch, Problem 10658: Another Type of Lattice Path, American Math. Monthly, 107, 2000, 368-370.

Cf. A027307, A108451, A108449, A108445.

A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3: g:=(1-z)/(1-z*(1-z)*A*(1+A)): gser:=series(g,z=0,24): 1,seq(coeff(gser,z^n),n=1..21);

{a(n)=local(y=1+x); for(i=1, n, y = -(-1 + 3*x - 3*x^2 + x^3 - 3*x^2*y + 2*x^3*y - 3*x*y^2 + 4*x^2*y^2 - 2*x^3*y^2 + x^4*y^2 - x*y^3 + 5*x^2*y^3 - 5*x^3*y^3 + 2*x^4*y^3) + (O(x^n))^4); polcoeff(y, n)}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Mar 18 2014

G.f.: (1-z)/[1-z(1-z)A(1+A)], where A=1+zA^2+zA^3=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3 (the g.f. of A027307).
From Vaclav Kotesovec, Mar 18 2014: (Start)
G.f. y(x) satisfies: -1 + 3*x - 3*x^2 + x^3 + y - 3*x^2*y + 2*x^3*y - 3*x*y^2 + 4*x^2*y^2 - 2*x^3*y^2 + x^4*y^2 - x*y^3 + 5*x^2*y^3 - 5*x^3*y^3 + 2*x^4*y^3 = 0
a(n) ~ (11+5*sqrt(5))^n * sqrt(247+603/sqrt(5)) / (5*sqrt(Pi)*n^(3/2) *2^(n+7/2))
Recurrence: n*(2*n + 1)*(6050*n^7 - 126115*n^6 + 1112432*n^5 - 5378320*n^4 + 15373805*n^3 - 25927435*n^2 + 23799813*n - 9117270)*a(n) = (193600*n^9 - 4138530*n^8 + 37940769*n^7 - 194878383*n^6 + 614482575*n^5 - 1224753180*n^4 + 1530842816*n^3 - 1150685847*n^2 + 475947900*n - 86751000)*a(n-1) - 2*(356950*n^9 - 7743285*n^8 + 72449748*n^7 - 382786506*n^6 + 1254763140*n^5 - 2635287165*n^4 + 3523007792*n^3 - 2857685139*n^2 + 1247080365*n - 211094100)*a(n-2) + (629200*n^9 - 13618110*n^8 + 127285773*n^7 - 672901416*n^6 + 2211415230*n^5 - 4666850055*n^4 + 6281980307*n^3 - 5134608429*n^2 + 2249815860*n - 375921000)*a(n-3) - (205700*n^9 - 4402860*n^8 + 40747203*n^7 - 213640971*n^6 + 697768275*n^5 - 1466844360*n^4 + 1971190342*n^3 - 1610202339*n^2 + 703447650*n - 115668000)*a(n-4) - 2*(n-5)*(2*n - 9)*(6050*n^7 - 83765*n^6 + 482792*n^5 - 1496135*n^4 + 2674295*n^3 - 2716295*n^2 + 1400898*n - 257040)*a(n-5)
(End)
D-finite with recurrence +n*(2*n+1)*(72425*n-317734)*a(n) +(-3140100*n^3+18675553*n^2-20491436*n+6673146)*a(n-1) +(22916600*n^3-190703953*n^2+432061605*n-302985732)*a(n-2) +2*(-37979850*n^3+409247558*n^2-1317355900*n+1324935945)*a(n-3) +3*(41724600*n^3-547102003*n^2+2263591341*n-2982348982)*a(n-4) +3*(-36023800*n^3+545643269*n^2-2684061391*n+4314486328)*a(n-5) +(46638250*n^3-790948395*n^2+4390868696*n-7976355570)*a(n-6) +(-6636700*n^3+127715416*n^2-812847607*n+1708833588)*a(n-7) -2*(266400*n-1297177)*(2*n-15)*(n-8)*a(n-8)=0. - R. J. Mathar, Jul 26 2022

A175124 A symmetric triangle, with sum the large Schröder numbers.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 10, 10, 1, 1, 20, 48, 20, 1, 1, 35, 161, 161, 35, 1, 1, 56, 434, 824, 434, 56, 1, 1, 84, 1008, 3186, 3186, 1008, 84, 1, 1, 120, 2100, 10152, 16840, 10152, 2100, 120, 1, 1, 165, 4026, 28050, 70807, 70807, 28050, 4026, 165, 1

Offset: 1

F. Chapoton, Feb 15 2010

a(n) is the number of noncrossing plants in the n+1 polygon, with no right corner, according to the number of left and top corners.
T(n,k) counts ordered complete binary trees with n leaves having k internal vertices colored black, the remaining n-1-k internal vertices colored white, and such that each vertex and its rightmost child have different colors. An example is given below. See Example 1.6.7 in [Drake] but note this triangle is not equal to A089447 as stated there. Compare with A196201. - Peter Bala, Sep 30 2011
Alternating sums seems to be A027307 (areated). - F. Chapoton, Mar 14 2024

			Triangle begins
n\k.|..1....2....3....4....5....6....7
= = = = = = = = = = = = = = = = = = = =
..1.|..1
..2.|..1....1
..3.|..1....4....1
..4.|..1...10...10....1
..5.|..1...20...48...20....1
..6.|..1...35..161..161...35....1
..7.|..1...56..434..824..434...56....1
...
Row 3: b^2+4*b*w+w^2. Internal vertices colored either b(lack) or w(hite); 3 uncolored leaf nodes shown as o.
.
  Weight     b^2           w^2
              b             w
             /\            /\
            /  \          /  \
           b    o        w    o
          /\            /\
         /  \          /  \
        o    o        o    o
.
  Weight       b*w
        b                   w
       /\                  /\
      /  \                /  \
     w    o              b    o
    /\                  /\
   /  \                /  \
  o    o              o    o
.
        b          w
       /\         /\
      /  \       /  \
     o    w     o    b
          /\         /\
         /  \       /  \
        o    o     o    o

Brian Drake, An inversion theorem for labeled trees and some limits of areas under lattice paths, A dissertation presented to the Faculty of the Graduate School of Arts and Sciences of Brandeis University.
Shishuo Fu, Z. Lin, and J. Zeng, Two new unimodal descent polynomials, arXiv preprint arXiv:1507.05184 [math.CO], 2015.
Robert Moerman and Lauren K. Williams, Grass(mannian) trees and forests: Variations of the exponential formula, with applications to the momentum amplituhedron, Comb. Theor. (2023) Vol. 3, No. 1, Art. 10, see p. 13.
Matteo Parisi, Melissa Sherman-Bennett, Ran Tessler, and Lauren Williams, The Magic Number Conjecture for the m=2 amplituhedron and Parke-Taylor identities, arXiv:2404.03026 [math.CO], 2024. See p. 8. See also Proc. 37th Conf. Formal Power Ser. Alg. Comb., Sém. Lotharingien Comb. (2025) Vol. 93B, Art. No. 103. See p. 5.
Matteo Parisi, Melissa Sherman-Bennett, and Lauren Williams, The m=2 amplituhedron and the hypersimplex: signs, clusters, triangulations, Eulerian numbers, arXiv:2104.08254 [math.CO], 2021.
Jeremy Quail and Puck Rombach, Positroid envelopes and graphic positroids, arXiv:2402.17841 [math.CO], 2024. See p. 31.

Cf. A006318 (row sums), A196201, A027307.

f:=RootOf((1+a*_Z)*(1+b*_Z)*x-_Z*(1-a*b*_Z^2));expand(taylor(f,x,4));

ab = InverseSeries[P*(1-a*b*P^2)/(1+a*P)/(1+b*P)+O[P]^12, P] // Normal // CoefficientList[#, P]&; (List @@@ ab) /. a|b -> 1 // Rest // Flatten (* Jean-François Alcover, Feb 23 2017 *)

G.f. is the composition inverse of P*(1-a*b*P^2)/(1+a*P)/(1+b*P).

A196201 T(n,k) counts ordered complete ternary trees with 2*n-1 leaves having k internal vertices colored black, the remaining n-1-k internal vertices colored white, and such that each vertex and its rightmost child have different colors.

Original entry on oeis.org

1, 1, 1, 2, 6, 2, 5, 28, 28, 5, 14, 120, 230, 120, 14, 27, 326, 985, 985, 326, 27, 56, 877, 3701, 5848, 3701, 877, 56, 116, 2212, 12096, 26988, 26988, 12096, 2212, 116, 221, 4808, 31740, 91402, 128738, 91402, 31740, 4808, 221

Offset: 1

Peter Bala, Sep 29 2011

Compare with Examples 1.6.7 and 1.6.9 in [Drake]. This triangle is a refinement of A027307. Compare with A175124.

			Triangle begins
n\k.|....1....2....3....4....5....6
= = = = = = = = = = = = = = = = = =
..1.|....1
..2.|....1....1
..3.|....2....6....2
..4.|....5...28...28....5
..5.|...14..120..230..120...14
..6.|...27..326..985..985..326...27
..
Row 3: 2*b^2+6*b*w+2w^2. Internal vertices colored either b(lack) or w(hite); 5 uncolored leaf nodes shown as o.
..Weights....b^2.......................w^2
........b...........b.............w...........w.....
......./|\........./|\.........../|\........./|\....
....../.|.\......./.|.\........./.|.\......./.|.\...
.....b..o..o.....o..b..o.......w..o..o.....o..w..o..
..../|\............/|\......../|\............/|\....
.../.|.\........../.|.\....../.|.\........../.|.\...
..o..o..o........o..o..o....o..o..o........o..o..o..
....................................................
..Weights....b*w..
........b...........b.............w...........w.....
......./|\........./|\.........../|\........./|\....
....../.|.\......./.|.\........./.|.\......./.|.\...
.....w..o..o.....o..w..o.......b..o..o.....o..b..o..
..../|\............/|\......../|\............/|\....
.../.|.\........../.|.\....../.|.\........../.|.\...
..o..o..o........o..o..o....o..o..o........o..o..o..
....................................................
........b...........w..........
......./|\........./|\.........
....../.|.\......./.|.\........
.....o..o..w.....o..o..b.......
........../|\........./|\......
........./.|.\......./.|.\.....
........o..o..o.....o..o..o....
...............................

B. Drake, An inversion theorem for labeled trees and some limits of areas under lattice paths, A dissertation presented to the Faculty of the Graduate School of Arts and Sciences of Brandeis University.

Cf. A027307 (row sums), A175124.

O.g.f.: compositional inverse of x-b*x^3/(1+b*x^2)-w*x^3/(1+w*x^2) = x +(b+w)*x^3 + (2*b^2+6*b*w+2*w^2)*x^5 + ....

A260776 Certain directed lattice paths.

Original entry on oeis.org

1, 5, 60, 1001, 19380, 408595, 9104550, 210905400, 5028168132, 122563178210, 3040594338320, 76520801509425, 1948777701739050, 50129821093933224, 1300611345665798320, 33994757853301868560, 894295618431497324900, 23660280955151412585930, 629143806228348421209768

Offset: 0

N. J. A. Sloane, Jul 30 2015

See Dziemianczuk (2014) for precise definition.

Lars Blomberg, Table of n, a(n) for n = 0..100
M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv preprint arXiv:1410.5747 [math.CO], 2014.

See Dziemianczuk (2014) Equation (36b) with N=3.
From Peter Bala, Mar 30 2020: (Start)
a(n) = (1/n)*C(5*n,2*n-1) for n >= 1.
a(n) = 2*Sum_{k = 0..n} C(3*n+k,n+3*k)*C(n+3*k,k)/(n+2*k+1). Cf. A027307(n) = Sum_{k = 0..n} C(2*n+k,n+2*k)*C(n+2*k,k)/(n+k+1) for n >= 1. (End)

More terms from Lars Blomberg, Aug 01 2015

A363418 Square array read by ascending antidiagonals: T(n,k) = [x^(n*k)] ((1 + x)/(1 - x))^k for n, k >= 1.

Original entry on oeis.org

2, 2, 8, 2, 16, 38, 2, 24, 146, 192, 2, 32, 326, 1408, 1002, 2, 40, 578, 4672, 14002, 5336, 2, 48, 902, 11008, 69002, 142000, 28814, 2, 56, 1298, 21440, 216002, 1038984, 1459810, 157184, 2, 64, 1766, 36992, 525002, 4320608, 15856206, 15158272, 864146

Offset: 1

Peter Bala, Jun 12 2023

The n-th row sequence {T(n, k) : k >= 1} satisfies the Gauss congruences, that is, T(n, m*p^r) == T(n, m*p^(r-1)) ( mod p^r ) for all primes p and positive integers m and r.

We conjecture that each row sequence satisfies the stronger supercongruences T(n, m*p^r) == T(n, m*p^(r-1)) ( mod p^(3*r) ) for all primes p >= 5 and positive integers m and r.

			Square array begins
 n\k |  1   2     3      4        5          6           7
 - - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  1  |  2   8    38    192     1002       5336       28814   ...   (A002003)
  2  |  2  16   146   1408    14002     142000     1459810   ...   (A103885)
  3  |  2  24   326   4672    69002    1038984    15856206   ...   (A333715)
  4  |  2  32   578  11008   216002    4320608    87588482   ...
  5  |  2  40   902  21440   525002   13104184   331482062   ...
  6  |  2  48  1298  36992  1086002   32497680   985524066   ...
  7  |  2  56  1766  58688  2009002   70097384  2478629134   ...
  8  |  2  64  2306  87552  3424002  136485568  5513464322   ...

R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)

A002003 (row 1), A103885 (row 2), A333715 (row 3). Cf. A035607, A362724 - A362733, A363419.

# display as a square array
T := (n,k) -> add( binomial(k, j)*binomial((n + 1)*k - j - 1, n*k - j) , j = 0..k): for n from 1 to 10 do seq(T(n, k), k = 1..10) end do;
#alternative program
seq(print(seq(simplify(2*k*hypergeom([1 - k, 1 - n*k], [2], 2)), k = 1..10)), n = 1..10);
# display as a sequence
seq(seq(T(n+1-i, i), i = 1..n), n = 1..10);

T(n,k) = sum(j=0, k, binomial(k, j)*binomial((n + 1)*k - j - 1, n*k - j)) \\ Andrew Howroyd, Jan 05 2024

T(n,k) = Sum_{j = 0..k} binomial(k, j)*binomial((n + 1)*k - j - 1, n*k - j).
T(n,k) = 1/n * [x^k] ((1 + x)/(1 - x))*(n*k).
T(n,k) = (1/n)*Sum_{j = 0..k} binomial(n*k, j)*binomial((n + 1)*k - j - 1, k - j).
T(2*n,k) = [x^(n*k)] Chebyshev_T(k,(1 + x)/(1 - x)), where Chebyshev_T(n,x) denotes the n-th Chebyshev polynomial of the first kind. See A053120.
T(n,k) = Sum_{j = 1..k} (2^j)*binomial(k, j)*binomial(n*k - 1, n*k - j).
T(n,k) = (2*k) * hypergeom([1 - k, 1 - n*k], [2], 2).
Define E(n,x) = exp( Sum_{j >= 1} T(n,j)*x^j/j ). Then T(n+1,k) = [x^k] E(n,x)^k.
E(n,x) = (1/x) * the series reversion of x/E(n-1,x) for n >= 2.
E(n,x)^n = (1/x) * the series reversion of x*((1 - x)/(1 + x))^n.
E(m,x) appears to be the g.f. of the (m + 1)-Schroeder numbers. See A027307 (m = 2) and the cross references there.
The o.g.f. for row n is the diagonal of the bivariate rational function (1/n) * t*f(x)^n/(1 - t*f(x)^n), where f(x) = (1 + x)/(1 - x), and hence is an algebraic function over Q(x) by Stanley 1999, Theorem 6.33, p. 197.

A102230 Triangle, read by rows, where each column equals the convolution of A032349 with the prior column, starting with column 0 equal to A032349 shift right.

Original entry on oeis.org

1, 1, 1, 4, 5, 1, 24, 32, 9, 1, 172, 236, 76, 13, 1, 1360, 1896, 656, 136, 17, 1, 11444, 16116, 5828, 1348, 212, 21, 1, 100520, 142544, 53112, 13184, 2376, 304, 25, 1, 911068, 1298524, 494364, 128924, 25436, 3804, 412, 29, 1, 8457504, 12100952

Offset: 0

Paul D. Hanna, Jan 01 2005

Row sums equal A027307; the self-convolution of the row sums form A032349. Column 0 equals A032349 shift right. Column 1 is A102231. This triangle is a variant of A100326.

			This triangle is generated by the recurrence:
T(n,k) = Sum_{i=0..n-k} T(i+1,0)*T(n-i-1,k-1) for n>k>0,
T(n,0) = Sum_{i=0..n-1} (2*i+1)*T(n-1,i) for n>0, with T(0,0)=1.
Rows begin:
[1],
[1,1],
[4,5,1],
[24,32,9,1],
[172,236,76,13,1],
[1360,1896,656,136,17,1],
[11444,16116,5828,1348,212,21,1],
[100520,142544,53112,13184,2376,304,25,1],...
Column 0 is formed from the partial sums of the prior row
after a term-by-term product with the odd numbers:
T(2,0) = 1*T(1,0) + 3*T(1,1) = 1*1 + 3*1 = 4.
T(3,0) = 1*T(2,0) + 3*T(2,1) + 5*T(2,2) = 1*4 + 3*5 + 5*1 = 24.

Cf. A032349, A027307, A102231, A100326.

PARI
```
{T(n,k)=if(n
				
```

G.f.: A(x, y) = (1+x*F(x))/(1-x*y*F(x)) where F(x) is the g.f. of A032349 and satisfies F(x) = (1+x*F(x))^2/(1-x*F(x))^2.

A108429 Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1) and have k down steps (d).

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 2, 5, 3, 0, 0, 0, 5, 21, 28, 12, 0, 0, 0, 0, 14, 84, 180, 165, 55, 0, 0, 0, 0, 0, 42, 330, 990, 1430, 1001, 273, 0, 0, 0, 0, 0, 0, 132, 1287, 5005, 10010, 10920, 6188, 1428, 0, 0, 0, 0, 0, 0, 0, 429, 5005, 24024, 61880, 92820, 81396, 38760, 7752, 0, 0, 0, 0

Offset: 0

Emeric Deutsch, Jun 03 2005

Row n contains 2n+1 terms, the first n of which are equal to 0.
Row sums yield A027307.
T(n,n) = A000108(n) (the Catalan numbers).
T(n,2n) = A001764(n) = binomial(3n,n)/(2n+1).
Except for the 0's, the same as A104978.
Number of d steps in all paths from (0,0) to (3n,0) is given by A108430.

			Example T(2,3) = 5 because we have udUdd, uUddd, Uddud, Ududd and Uuddd.
Triangle begins:
1;
0,1,1;
0,0,2,5,3;
0,0,0,5,21,28,12;
...

Emeric Deutsch, Problem 10658: Another Type of Lattice Path, American Math. Monthly, 107, 2000, 368-370.

Cf. A027307, A000108, A001764, A104978, A108430.

a:=proc(n,k) if n=0 and k=0 then 1 elif n=0 then 0 elif k=0 then 0 else binomial(n,2*n-k)*binomial(n+k,n-1)/n fi end: for n from 0 to 8 do seq(a(n,k),k=0..2*n) od; # yields sequence in triangular form

T(n,k) = binomial(n,2n-k)*binomial(n+k, n-1)/n.

G.f.: G = G(t, z) satisfies G=1+tzG^2*(1+tG).

A108430 Number of d steps in all paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1).

Original entry on oeis.org

3, 31, 311, 3151, 32299, 334335, 3488239, 36627487, 386618387, 4098713631, 43611791783, 465496885231, 4981942135611, 53443871159551, 574500093677535, 6186886528903231, 66735614131858723, 720897596248427295

Offset: 1

Emeric Deutsch, Jun 03 2005

nonn

			a(1) = 3 because in the paths ud, Udd we have 3 d steps altogether.

Emeric Deutsch, Problem 10658: Another Type of Lattice Path, American Math. Monthly, 107, 2000, 368-370.

Cf. A027307, A108429.

a:=n->(1/n)*sum(k*binomial(n,2*n-k)*binomial(n+k,n-1),k=n..2*n): seq(a(n),n=1..22);

a(n) = (1/n)*sum(k*binomial(n,2n-k)*binomial(n+k,n-1), k=n..2n).

Conjecture D-finite with recurrence n*(2*n+1)*(23982*n-28681)*a(n) +(-640736*n^3+1168048*n^2-901220*n+247035)*a(n-1) +(1196488*n^3-6448608*n^2+10992587*n-5911365)*a(n-2) +2*(2*n-5)*(28283*n-6993)*(n-3)*a(n-3)=0. - R. J. Mathar, Jul 24 2022

A108436 Number of returns to the x-axis in all paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1).

Original entry on oeis.org

2, 14, 106, 862, 7378, 65550, 599002, 5594942, 53181730, 512784142, 5003410762, 49312114334, 490192537586, 4909102791694, 49482525122490, 501626536004734, 5111038278845506, 52312236295906830, 537605889306476074

Offset: 1

Emeric Deutsch, Jun 04 2005

nonn

			a(2)=14 because there are 10 paths from (0,0) to (6,0) (see A027307): u(d)u(d), u(d)Ud(d), uud(d), uUdd(d), Ud(d)u(d), Ud(d)Ud(d), Udud(d), UdUdd(d), Uudd(d) and UUddd(d), the fourteen returns to the x-axis being shown between parentheses.

Emeric Deutsch, Problem 10658: Another Type of Lattice Path, American Math. Monthly, 107, 2000, 368-370.

Cf. A027307, A108435, A032349.

a:=n->2+(1/n)*sum((3*n-j)*2^(n-j)*binomial(n,j)*binomial(2*n,n-j-1)/(n+j+2),j=0..n-2): seq(a(n),n=1..21);

a(n) = Sum_{k=1..n} k*A108435(k).
a(n) = A032349(n+1) - A027307(n).
a(n) = 2 + (1/n)*Sum_{j=0..n-2} (3n-j)*2^(n-j)*binomial(n, j)*binomial(2n, n-j-1)/(n+j+2).
G.f.: A^2-A, where A=1+zA^2+zA^3 or, equivalently, A=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3 (the g.f. of A027307).
D-finite with recurrence 3*(n+1)*(2*n+1)*a(n) +3*(-34*n^2+18*n-1)*a(n-1) +(394*n^2-1197*n+908)*a(n-2) +2*(-4*n^2+51*n-113)*a(n-3) -2*(2*n-7)*(n-4)*a(n-4)=0. - R. J. Mathar, Jul 24 2022

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A196201 T(n,k) counts ordered complete ternary trees with 2*n-1 leaves having k internal vertices colored black, the remaining n-1-k internal vertices colored white, and such that each vertex and its rightmost child have different colors.

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A260776 Certain directed lattice paths.

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A363418 Square array read by ascending antidiagonals: T(n,k) = [x^(n*k)] ((1 + x)/(1 - x))^k for n, k >= 1.

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A102230 Triangle, read by rows, where each column equals the convolution of A032349 with the prior column, starting with column 0 equal to A032349 shift right.

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A108429 Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1) and have k down steps (d).

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