A027307 -id:A027307 - cp's OEIS frontend

A108447 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 no peaks of the form ud.

1, 1, 4, 20, 113, 688, 4404, 29219, 199140, 1385904, 9807820, 70364704, 510609620, 3741212535, 27639233548, 205660399220, 1539916433473, 11594310041792, 87725707127600, 666681174728724, 5086601816592432, 38948589882247968

Offset: 0

Emeric Deutsch, Jun 10 2005

nonn

Column 0 of A108446.

			a(2)=4 because we have uUddd, UddUdd, UdUddd and UUdddd.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Emeric Deutsch, Problem 10658: Another Type of Lattice Path, American Math. Monthly, 107, 2000, 368-370.

Cf. A000108, A027307, A108446, A108425, A108426.

a:=n->(1/n)*sum(binomial(n,j)*binomial(n+2*j,j-1),j=0..n): 1, seq(a(n),n=1..25);
a := n -> `if`(n=0,1,simplify(hypergeom([1-n,(n+3)/2,(n+4)/2],[2, n+3],-4))): seq(a(n), n=0..21); # Peter Luschny, Oct 30 2015

Flatten[{1, Table[Sum[Binomial[n, j]*Binomial[n + 2*j, j-1], {j, 0, n}]/n, {n, 1, 20}]}] (* Vaclav Kotesovec, Nov 27 2017 *)
terms = 22; g[] = 1; Do[g[x] = 1+x*g[x]*(g[x]^2+g[x]-1) + O[x]^terms // Normal, {terms}]; CoefficientList[g[x], x] (* Jean-François Alcover, Jul 19 2018 *)

a(n) = (1/n) * Sum_{j=0..n} binomial(n, j)*binomial(n+2j, j-1) (n>=1); a(0)=1.
G.f.: G satisfies G = 1 + z*G*(G^2+G-1).
a(n) = hypergeom([1-n,(n+3)/2,(n+4)/2],[2,n+3],-4) for n>=1. - Peter Luschny, Oct 30 2015
a(n) ~ sqrt((s-1) / (Pi*(1 + 3*s))) / (2*n^(3/2) * r^(n + 1/2)), where r = 0.1215851068721183026145063923222031450327682505108... and s = 1.451605962955776643742608112028547116887657025022... are real roots of the system of equations 1 + r*s*(-1 + s + s^2) = s, r*(-1 + 2*s + 3*s^2) = 1. - Vaclav Kotesovec, Nov 27 2017
O.g.f.: A(x) = (1/x) * Revert( x/c(x/(1 - x)) ), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. of the Catalan numbers A000108. - Peter Bala, Mar 08 2020
D-finite with recurrence 8*n*(2*n+1)*a(n) -6*(2*n-1)*(13*n-10)*a(n-1) +24*(4*n-7)*(2*n-5)*a(n-2) +4*(19*n-40)*(n-3)*a(n-3) -35*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Jul 26 2022

A260332 Labelings of n diamond-shaped posets with 4 vertices per diamond where the labels follow the poset relations whose associated reading permutation avoids 231 in the classical sense.

Original entry on oeis.org

1, 2, 18, 226, 3298, 52450, 881970

Offset: 0

Manda Riehl, Jul 29 2015

According to Yang-Jiang (2021) these are the 5-Schroeder numbers. If confirmed, this will prove Michael Weiner's conjectures and enable us to extend the sequence. Yang & Jiang (2021) give an explicit formula for the m-Schroeder numbers in Theorem 2.4. - N. J. A. Sloane, Mar 28 2021
By diamond-shaped poset with 4 vertices, we mean a poset on four elements with e_1 < e_2, e_1 < e_3, e_2 < e_4, e_3 < e_4, and no order relations between e_2 and e_3. In the Hasse diagram for such a poset, we have a least element, two elements in the level above, and one element in the top level, so the diagram resembles a diamond.  The associated permutation is the permutation obtained by reading the labels of each poset by levels left to right, starting with the least element.
Also the number of labelings of n diamond-shaped posets with 4 vertices per diamond where the labels follow the poset relations whose associated reading permutation avoids 312 in the classical sense via reverse complement Wilf equivalence.
Conjecture: Also the number of lattice paths (Schroeder paths) from (0,0) to (n,4n) with unit steps N=(0,1), E=(1,0) and D=(1,1) staying weakly above the line y = 4x. - Michael D. Weiner, Jul 24 2019

			For a single diamond (n=1) poset with 4 vertices, we must label the least element 1 and the greatest element 4, and the two central elements can be labeled either 2, 3 or 3, 2 respectively. Thus the associated permutations are 1234 and 1324.

Sheng-Liang Yang and Mei-yang Jiang, The m-Schröder paths and m-Schröder numbers, Disc. Math. (2021) Vol. 344, Issue 2, 112209. doi:10.1016/j.disc.2020.112209. See Table 1.

M. Paukner, L. Pepin, M. Riehl, and J. Wieser, Pattern Avoidance in Task-Precedence Posets, arXiv:1511.00080 [math.CO], 2015.
Manda Riehl, A 231-avoiding diamond whose associated permutation is 1234.
Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 11.
Lin Yang, Yu-Yuan Zhang, and Sheng-Liang Yang, The halves of Delannoy matrix and Chung-Feller properties of the m-Schröder paths, Linear Alg. Appl. (2024).
Sheng-liang Yang and Mei-yang Jiang, Pattern avoiding problems on the hybrid d-trees, J. Lanzhou Univ. Tech., (China, 2023) Vol. 49, No. 2, 144-150. (in Mandarin)

Cf. A260331, A260579.

The sequences listed in Yang-Jiang's Table 1 appear to be A006318, A001003, A027307, A034015, A144097, A243675, A260332, A243676. - N. J. A. Sloane, Mar 28 2021

There is a complicated recursive formula available in Paukner et al.
Yang & Jiang (2021) give an explicit formula for the 5-Schroeder numbers in Theorem 2.4. - N. J. A. Sloane, Mar 28 2021
Conjecture: a(n) = Sum_{k=1..n} binomial(n,k)*binomial(4*n,k-1)*2^k/n for n > 0. - Michael D. Weiner, Jul 23 2019
From Peter Bala, Jun 16 2023: (Start)
Conjectures: 1) the g.f. A(x) = 1 + 2*x + 18*x^2 + 226*x^3 + ... satisfies A(x)^4 = (1/x) * the series reversion of ((1 - x)/(1 + x))^4.
2) Define b(n) = (1/4) * [x^n] ((1 + x)/(1 - x))^(4*n). Then A(x) = exp( Sum_{n >= 1} b(n)*x^n/n ).
3) a(n) = 2 * hypergeom([1 - n, -4*n], [2], 2) for n >= 1 (equivalent to Weiner's conjecture above).
4) [x^n] A(x)^n =  (2*n) * hypergeom([1 - n, 1 - 5*n], [2], 2) for n >= 1. (End)

A333090 a(n) is equal to the n-th order Taylor polynomial (centered at 0) of S(x)^n evaluated at x = 1, where S(x) = (1 - x - sqrt(1 - 6x + x^2))/(2x) is the o.g.f. of the Schröder numbers A006318.

Original entry on oeis.org

1, 3, 21, 183, 1729, 17003, 171237, 1752047, 18130433, 189218451, 1987916021, 20996253479, 222730436161, 2371369720827, 25325636818629, 271189884041183, 2910628489408513, 31302328583021091, 337241582882175189, 3639109029230457751, 39324814984207649729

Offset: 0

Peter Bala, Mar 22 2020

The sequence satisfies the Gauss congruences: a(n*p^k) == a(n*p^(k-1)) ( mod p^k ) for all prime p and positive integers n and k.
We conjecture that the sequence satisfies the stronger supercongruences a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for all primes p >= 5 and positive integers n and k. Examples of these congruences are given below.
More generally, for each integer m, we conjecture that the sequence {a_m(n) : n >= 0}, defined by setting a_m(n) = the n-th order Taylor polynomial of S(x)^(m*n) evaluated at x = 1, satisfies the same supercongruences. See A333091 for m = 2 and A333092 for m = 3. For similarly defined sequences see A333093 through A333097.

			n-th order Taylor polynomial of S(x)^n:
  n = 0: S(x)^0 = 1 + O(x)
  n = 1: S(x)^1 = 1 + 2*x + O(x^2)
  n = 2: S(x)^2 = 1 + 4*x + 16*x^2 + O(x^3)
  n = 3: S(x)^3 = 1 + 6*x + 30*x^2 + 146*x^3 + O(x^4)
  n = 4: S(x)^4 = 1 + 8*x + 48*x^2 + 264*x^3 + 1408*x^4 + O(x^5)
Setting x = 1 gives a(0) = 1, a(1) = 1 + 2 = 3, a(2) = 1 + 4 + 16 = 21, a(3) = 1 + 6 + 30 + 146 = 183 and a(4) = 1 + 8 + 48 + 264 + 1408 = 1729.
The triangle of coefficients of the n-th order Taylor polynomial of S(x)^n, n >= 0, in descending powers of x begins
                                          row sums
  n = 0 |    1                                1
  n = 1 |    2    1                           3
  n = 2 |   16    4    1                     21
  n = 3 |  146   30    6   1                183
  n = 4 | 1408  264   48   8   1           1729
   ...
This is a Riordan array belonging to the Hitting time subgroup of the Riordan group. The first column sequence is [x^n]S(x)^n = A103885(n).
Examples of supercongruences:
a(13) - a(1) = 2371369720827 - 3 = (2^3)*(3^2)*(13^3)*83*180617 == 0 ( mod 13^3 ).
a(3*7) - a(3) = 425495386400395896971 - 183 = (2^2)*(7^3*)*19*47* 347287606554703 == 0 ( mod 7^3 ).
a(5^2) - a(5) = 5894174066435445232142003 - 17003 = (2^3)*(3^4)*(5^6)*17* 41*101*5081*1627513421 == 0 ( mod 5^6 ).

Cf. A006318, A027307, A103885, A333091 through A333097, A372214, A372215.

S:= x -> (1/2)*(1-x-sqrt(1-6*x+x^2))/x:
G := (x,n) -> series(S(x)^n, x, 51):
seq(add(coeff(G(x, n), x, k), k = 0..n), n = 0..25);

Table[SeriesCoefficient[((1 + x)*(1 - Sqrt[1 - 4*x - 4*x^2])/(2*x))^n, {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 28 2020 *)

a(n) = [x^n] ( (1 + x)*S(x/(1 + x)) )^n.
O.g.f.: ( 1 + x*f'(x)/f(x) )/( 1 - x*f(x) ), where f(x) = 1 + 2*x + 10*x^2 + 66*x^3 + 498*x^4 + ... = (1/x)*Revert( x/S(x) ) is the o.g.f. of A027307.
Row sums of the Riordan array ( 1 + x*f'(x)/f(x), x*f(x) ) belonging to the Hitting time subgroup of the Riordan group.
a(n) ~ phi^(5*n+2) / (2*5^(3/4)*sqrt(Pi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Mar 28 2020

A026000 a(n) = T(2n, n), where T is the Delannoy triangle (A008288).

Original entry on oeis.org

1, 5, 41, 377, 3649, 36365, 369305, 3800305, 39490049, 413442773, 4354393801, 46082942185, 489658242241, 5220495115997, 55818956905529, 598318746037217, 6427269150511105, 69175175263888037, 745778857519239785, 8052432236270744665, 87063177396677721409

Offset: 0

Clark Kimberling

Even order terms in the diagonal of rational function 1/(1 - (x + y^2 + x*y^2)). - Gheorghe Coserea, Aug 31 2018

			A(x) = 1 + 5*x + 41*x^2 + 377*x^3 + 3649*x^4 + 36365*x^5 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Lin Yang, Yu-Yuan Zhang, and Sheng-Liang Yang, The halves of Delannoy matrix and Chung-Feller properties of the m-Schröder paths, Linear Alg. Appl. (2024).

Cf. A008288, A027307, A026001.

Flatten[{1,RecurrenceTable[{2*n*(2*n-1)*a[n] == (46*n^2-51*n+15)*a[n-1] - (18*n^2-82*n+85)*a[n-2] - (n-2)*(2*n-5)*a[n-3],a[1]==5,a[2]==41,a[3]==377},a,{n,20}]}] (* Vaclav Kotesovec, Oct 08 2012 *)
a[n_] :=  HypergeometricPFQ[{-n, -n, n + 1}, {1/2, 1}, 1];
Table[a[n], {n, 0, 18}] (* Peter Luschny, Mar 14 2018 *)

seq(N) = {
  my(a = vector(N)); a[1]=5; a[2]=41; a[3]=377;
  for (n=4, N,
    a[n] = (46*n^2-51*n+15)*a[n-1] - (18*n^2-82*n+85)*a[n-2] - (n-2)*(2*n-5)*a[n-3];
    a[n] /= 2*n*(2*n-1));
  concat(1, a);
};
seq(18)
\\ test: y=Ser(seq(303),'x); 0 == 4*(x^2 + 11*x - 1)*y^3 + (x + 3)*y + 1
\\ Gheorghe Coserea, Aug 31 2018

a(n) = ((2*n+3)*(n+1)*A027307(n+1)/2-(3*n+2)*n*A027307(n)) / (5*n+3) (guessed). - Mark van Hoeij, Jul 02 2010
Recurrence: 2*n*(2*n-1)*a(n) = (46*n^2-51*n+15)*a(n-1) - (18*n^2-82*n+85)*a(n-2) - (n-2)*(2*n-5)*a(n-3). - Vaclav Kotesovec, Oct 08 2012
a(n) ~ sqrt(150+70*sqrt(5))*((11+5*sqrt(5))/2)^n/(20*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 08 2012. Equivalently, a(n) ~ phi^(5*n + 2) / (2 * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 08 2021
a(n) = hypergeom([-n, -n, n + 1], [1/2,  1], 1). - Peter Luschny, Mar 14 2018
From Gheorghe Coserea, Aug 31 2018:(Start)
G.f.: 1 + serreverse((-(44*x^2 + 88*x + 45) + (10*x + 9)*sqrt(20*x^2 + 44*x + 25))/(8*(x + 1)^2)).
G.f. y=A(x) satisfies:
0 = 4*(x^2 + 11*x - 1)*y^3 + (x + 3)*y + 1.
0 = 2*x*(x - 2)*(x^2 + 11*x - 1)*y'' + (5*x^3 + 8*x^2 - 87*x + 2)*y' + (x^2 - 7*x - 10)*y. (End)
From Peter Bala, Jan 20 2020: (Start)
a(n) = Sum_{k = 0..n} C(2*n, n-k) * C(2*n+k, k).
a(n) = C(2*n, n) * hypergeom([-n, 2*n+1], [n+1], -1).
n*(2*n-1)*(10*n-13)*a(n) = (220*n^3-506*n^2+334*n-63*n)*a(n-1) + (n-1)*(2*n-3)*(10*n-3)*a(n-2). (End)
From Peter Bala, Apr 15 2023: (Start)
a(n) = Sum_{k = 0..n} binomial(n, k)*binomial(2*n, k)*2^k
a(n) = (-1)^n * Sum_{k = 0..n} binomial(n, k)*binomial(2*n+k, k)*(-2)^k.
a(n) = hypergeom([-n, -2*n], [1], 2) = (-1)^n * hypergeom([-n, 2*n + 1], [1], 2). (End)

A363311 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x)^3 + A(x)^5).

Original entry on oeis.org

1, 2, 16, 180, 2360, 33760, 510928, 8043440, 130371936, 2161066432, 36465401344, 624274702464, 10816259970048, 189305983870208, 3341924242051840, 59437975940616960, 1064030847809734144, 19157066319365860352, 346663014660754833408, 6301645517153393121280

Offset: 0

Paul D. Hanna, May 29 2023

nonn

			G.f.: A(x) = 1 + 2*x + 16*x^2 + 180*x^3 + 2360*x^4 + 33760*x^5 + 510928*x^6 + 8043440*x^7 + 130371936*x^8 + 2161066432*x^9 + 36465401344*x^10 + ...
where A(x) = 1 + x*(A(x)^3 + A(x)^5).
RELATED SERIES.
A(x)^3 = 1 + 6*x + 60*x^2 + 740*x^3 + 10200*x^4 + 150576*x^5 + 2328640*x^6 + 37242096*x^7 + ...
A(x)^5 = 1 + 10*x + 120*x^2 + 1620*x^3 + 23560*x^4 + 360352*x^5 + 5714800*x^6 + 93129840*x^7 + ... + A363310(n-1)*x^n + ...

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A363309, A363310, A027307, A001764.

A363311 := proc(n)
    add(binomial(n,k)*binomial(3*n+2*k+1,n)/(3*n+2*k+1),k=0..n) ;
end proc:
seq(A363311(n),n=0..70) ; # R. J. Mathar, Jul 18 2023

{a(n) = sum(k=0, n, binomial(n, k)*binomial(3*n+2*k+1, n)/(3*n+2*k+1) )}
for(n=0, 20, print1(a(n), ", "))

G.f.: A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) A(x) = 1 + x*(A(x)^3 + A(x)^5).
(2) A(x) = ((B(x) - 1)/x)^(1/5) where B(x) is the g.f. of A363310.
(3) a(n) = Sum_{k=0..n} binomial(n, k)*binomial(3*n+2*k+1, n)/(3*n+2*k+1) for n >= 0.
D-finite with recurrence +8*n*(9639909229907389*n -4332180801077160)* (4*n+1) *(2*n-1) *(4*n-1) *a(n) +(-76286895522125418545*n^5 +381775644252842912682*n^4 -1033993649015194853931*n^3 +1551245138730960078498*n^2 -1139936487176542639744*n +315922393907140666080) *a(n-1) +2*(272671960126472445261*n^5 -3010900995907383509536*n^4 +12907236726784549786263*n^3 -27012522362058892089464*n^2 +27708850835094249342996*n -11174516509692301247280) *a(n-2) +4*(-627566489435411923*n^5 +144061968293307107646*n^4 -1706290600068411299693*n^3 +7720188970563268791354*n^2 -15561118085635458987024*n +11755034318370549299520) *a(n-3) -8*(n-4) *(696748847001815555*n^4 -19100265029551686306*n^3 +142472091583377235329*n^2 -415309555491080054458*n +422902881832258952040) *a(n-4) -96*(n-4) *(n-5)*(3*n-13) *(2465432947213573*n -7363340799047272) *(3*n-14) *a(n-5)=0. - R. J. Mathar, Jul 18 2023
a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} 2^(n-k) * binomial(n,k) * binomial(4*n-k,n-1-2*k) for n > 0. - Seiichi Manyama, Apr 01 2024

A085403 Expansion of (1-x+sqrt(1-6x+x^2))/2 in powers of x.

Original entry on oeis.org

1, -2, -2, -6, -22, -90, -394, -1806, -8558, -41586, -206098, -1037718, -5293446, -27297738, -142078746, -745387038, -3937603038, -20927156706, -111818026018, -600318853926, -3236724317174, -17518619320890, -95149655201962, -518431875418926, -2832923350929742

Offset: 0

Michael Somos, Jun 28 2003

sign

Series reversion of x(Sum_{k>=0} a(k)x^k) is x(Sum_{k>=0} A027307(k)x^k).

Vincenzo Librandi, Table of n, a(n) for n = 0..200

A minor variation of A006318. a(n)=-A006318(n-1), n>1.

CoefficientList[Series[(1-x+Sqrt[1-6x+x^2])/2,{x,0,30}],x] (* Harvey P. Dale, Jun 13 2013 *)

a(n)=polcoeff((1-x+sqrt(1-6*x+x^2+x*O(x^n)))/2,n)

G.f.: (1-x+sqrt(1-6x+x^2))/2. (=1/g.f. A006318)
Given g.f. A(x), y=A(x)x satisfies 0=f(x, y) where f(x, y)=y(y-x)+(x+y)x^2 . - Michael Somos, May 23 2005
G.f.: Q(0) where Q(k) = 1 + k*(1-x) - x - x*(k+1)*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Mar 14 2013
a(n) ~ -sqrt(3*sqrt(2)-4) * (3+2*sqrt(2))^n / (2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 20 2014
D-finite with recurrence: n*a(n) +3*(-2*n+3)*a(n-1) +(n-3)*a(n-2)=0. - R. J. Mathar, Jan 20 2020

A104978 Triangle read by rows, where the g.f. satisfies A(x, y) = 1 + xA(x, y)^2 + xy*A(x, y)^3.

Original entry on oeis.org

1, 1, 1, 2, 5, 3, 5, 21, 28, 12, 14, 84, 180, 165, 55, 42, 330, 990, 1430, 1001, 273, 132, 1287, 5005, 10010, 10920, 6188, 1428, 429, 5005, 24024, 61880, 92820, 81396, 38760, 7752, 1430, 19448, 111384, 352716, 678300, 813960, 596904, 245157, 43263, 4862, 75582, 503880, 1899240, 4476780, 6864396, 6864396, 4326300, 1562275, 246675

Offset: 0

Paul D. Hanna, Mar 30 2005

			The triangle T(n, k) begins:
  [0]    1;
  [1]    1,     1;
  [2]    2,     5,      3;
  [3]    5,    21,     28,     12;
  [4]   14,    84,    180,    165,     55;
  [5]   42,   330,    990,   1430,   1001,    273;
  [6]  132,  1287,   5005,  10010,  10920,   6188,   1428;
  [7]  429,  5005,  24024,  61880,  92820,  81396,  38760,   7752;
  [8] 1430, 19448, 111384, 352716, 678300, 813960, 596904, 245157, 43263;
  ...
The array A(n, k) begins:
  [0]   1,    1,      3,      12,       55,       273,       1428, ...  [A001764]
  [1]   1,    5,     28,     165,     1001,      6188,      38760, ...  [A025174]
  [2]   2,   21,    180,    1430,    10920,     81396,     596904, ...  [A383450]
  [3]   5,   84,    990,   10010,    92820,    813960,    6864396, ...  [A383451]
  [4]  14,  330,   5005,   61880,   678300,   6864396,   65615550, ...
  [5]  42, 1287,  24024,  352716,  4476780,  51482970,  551170620, ...
  [6] 132, 5005, 111384, 1899240, 27457584, 354323970, 4206302100, ...
  [A000108]  |  [A074922][A383452]
         [A002054]

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
N. J. Wildberger and Dean Rubine, A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, Amer. Math. Monthly (2025). See sections 8 and 12.
Jian Zhou, Fat and Thin Emergent Geometries of Hermitian One-Matrix Models, arXiv:1810.03883 [math-ph], 2018.

Columns of array: A000108, A002054, A074922, A383452.
Rows of array: A001764, A025174, A383450, A383451.
Cf. A001002 (antidiagonal sums), A001764 (semidiagonal sums), A027307 (row sums), A104979, A383439 (central terms).

[Binomial(2*n+k, n+2*k)*Binomial(n+2*k, k)/(n+k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 08 2021

From Peter Luschny, May 04 2025:  (Start)
T := (n, k) -> (k + 2*n)!/(k!*(n - k)!*(n + k + 1)!):
seq(print(seq(T(n, k), k = 0..n)), n = 0..10);
# Alternatively the array:
A := (n, k) -> (3*k + 2*n)!/(k!*n!*(n + 2*k + 1)!);
for n from 0 to 8 do seq(A(n, k), k = 0..7) od;  (End)

T[n_, k_]:= Binomial[2n+k, n+2k]*Binomial[n+2k, k]/(n+k+1);
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* Jean-François Alcover, Jan 27 2019 *)

T(n,k) = local(A=1+x+x*y+x*O(x^n)+y*O(y^k)); for(i=1,n,A=1+x*A^2+x*y*A^3); polcoeff(polcoeff(A,n,x),k,y)
for(n=0, 10, for(k=0, n, print1(T(n,k),", ")); print(""))

Dy(n, F)=local(D=F); for(i=1, n, D=deriv(D,y)); D
T(n,k)=local(A=1); A=1+sum(m=1, n+1, x^m/y^(m+1) * Dy(m-1, (y^2+y^3)^m/m!)) +x*O(x^n)+y*O(y^k); polcoeff(polcoeff(A, n,x),k,y)
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print()) \\ Paul D. Hanna, Jun 22 2012

x='x; y='y; z='z; Fxyz = 1 - z + x*z^2 + x*y*z^3;
seq(N) = {
  my(z0 = 1 + O((x*y)^N), z1 = 0);
  for (k = 1, N^2,
    z1 = z0 - subst(Fxyz, z, z0)/subst(deriv(Fxyz, z), z, z0);
    if (z0 == z1, break()); z0 = z1);
  vector(N, n, Vecrev(polcoeff(z0, n-1, 'x)));
};
concat(seq(9)) \\ Gheorghe Coserea, Nov 30 2016

flatten([[binomial(2*n+k, n+2*k)*binomial(n+2*k, k)/(n+k+1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 08 2021

T(n, k) = binomial(2*n+k, n+2*k)*binomial(n+2*k, k)/(n+k+1).
G.f.: A(x, y) = Sum_{n>=0} x^n/y^(n+1) * d^(n-1)/dy^(n-1) (y^2 + y^3)^n / n!. - Paul D. Hanna, Jun 22 2012
G.f. of row n: 1/y^(n+1) * d^(n-1)/dy^(n-1) (y^2+y^3)^n / n!. - Paul D. Hanna, Jun 22 2012
A(n, k) = T(n + k, k) = (3*k + 2*n)! / (k!*n!*(n + 2*k + 1)!). - Peter Luschny, May 04 2025

A243675 Number of hypoplactic classes of 3-parking functions of length n.

Original entry on oeis.org

1, 1, 7, 67, 741, 8909, 113107, 1492103, 20251945, 280978681, 3967031839, 56811348235, 823250855181, 12049087175493, 177857857845675, 2644773866954255, 39581787842355409, 595745692419162737, 9011736489133233463, 136932249972928786387, 2089082351509217490613

Offset: 0

N. J. A. Sloane, Jun 14 2014

nonn

See Novelli-Thibon (2014) for precise definition.
This is almost certainly the sequence of small 4-Schroeder numbers as defined by Yang-Jiang (2021). It would be nice to have a proof. Then we could confirm Weiner's conjectured formula, and extend the sequence. Yang & Jiang (2021) give an explicit formula for the small m-Schroeder numbers in Theorems 2.4 and 2.9. - N. J. A. Sloane, Mar 28 2021
This is also the small 4-Schroeder numbers defined by Yang and Jiang (2021) in Theorems 2.4 and 2.9. - Jun Yan, Apr 13 2024

Sheng-Liang Yang and Mei-yang Jiang, The m-Schröder paths and m-Schröder numbers, Disc. Math. (2021) Vol. 344, Issue 2, 112209. doi:10.1016/j.disc.2020.112209. See Table 1.

J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014-2020. See Fig. 23.
Jun Yan, Results on pattern avoidance in parking functions, arXiv preprint arXiv:2404.07958 [math.CO], 2024. See Theorem 4.4.

The sequences listed in Yang-Jiang's Table 1 appear to be A006318, A001003, A027307, A034015, A144097, A243675, A260332, A243676. - N. J. A. Sloane, Mar 28 2021

Apparently, a(n) = A144097/2, apart from the initial term. - N. J. A. Sloane, Mar 28 2021 [This is for n > 0 indeed the case. - Jun Yan, Apr 13 2024]

a := proc(n) option remember; if n <= 1 then return 1 fi;
-((945*n^5 - 5481*n^4 + 11685*n^3 - 11091*n^2 + 4470*n - 600)*a(n - 2) +
(-15610*n^5 + 67123*n^4 - 106824*n^3 + 77633*n^2 - 25514*n + 3000)*a(n - 1)) /
(945*n^5 - 2646*n^4 + 1731*n^3 + 294*n^2 - 204*n) end:
seq(a(n), n = 0..20);  # Peter Luschny, Apr 13 2024

a[n_] := Hypergeometric2F1[1 - n, -3 n, 2, 2];
Table[a[n], {n, 0, 20}]  (* Peter Luschny, Apr 13 2024 *)

a(n) = Sum_{i=0..n} Sum_{j=0..i} (-2)^(n-i)*binomial(i,j)*binomial(3i+j, n)*binomial(n+1,i)/(n+1) (conjectured). - Michael D. Weiner, May 25 2017
From Jun Yan, Apr 13 2024 : (Start)
a(n) = (1/n) * Sum_{k=1..n} binomial(3*n, k - 1) * binomial(n, k)*2^(k - 1) for n>0.
Let D(n) be the set of 3-Dyck paths with n up-steps of size 3, 3n down-steps of size 1 and never go below the x-axis. For every d in D(n), let peak(d) be the number of peaks in d. Then a(n) = Sum_{d in D(n)}2^{peak(d) - 1}. (End)
a(n) = hypergeom([1 - n, -3*n], [2], 2). - Peter Luschny, Apr 13 2024
D-finite with recurrence -15*n*(3*n-1)*(3*n+1)*a(n) +(43*n^3+5403*n^2-8482*n+3228)*a(n-1) +6*(6039*n^3-33372*n^2+60401*n-35858) *a(n-2) +9*(-689*n^3+5938*n^2-17157*n+16616)*a(n-3) +27*(3*n-10)*(3*n-11)*(n-4)*a(n-4)=0. - R. J. Mathar, Apr 14 2024
D-finite with recurrence 3*n*(3*n-1)*(3*n+1)*(35*n^2-98*n+68)*a(n) +(-15610*n^5+67123*n^4-106824*n^3+77633*n^2-25514*n+3000)*a(n-1) +3*(n-2)*(3*n-4)*(3*n-5)*(35*n^2-28*n+5)*a(n-2)=0. - R. J. Mathar, Apr 14 2024

Added a(0) = 1. - N. J. A. Sloane, Mar 28 2021

More terms from Jun Yan, Apr 13 2024

A243676 Number of hypoplactic classes of 4-parking functions of length n.

Original entry on oeis.org

1, 1, 9, 113, 1649, 26225, 440985, 7711009, 138792929, 2554489505, 47854963881, 909495557393, 17492724268369, 339846019830673, 6659441891042105, 131467175048437569, 2612224160086781889, 52201209713045788737, 1048450942860766632777, 21153308764742204273329, 428520989167282737342513

Offset: 0

N. J. A. Sloane, Jun 14 2014

nonn

This is almost certainly the sequence of small 5-Schroeder numbers as defined by Yang-Jiang (2021). It would be nice to have a proof. Then we could confirm Weiner's conjectured formulas, and extend the sequence. Yang & Jiang (2021) give an explicit formula for the small m-Schroeder numbers in Theorems 2.4 and 2.9. - N. J. A. Sloane, Mar 28 2021

This is indeed the small 5-Schroeder numbers defined by Yang and Jiang (2021) in Theorems 2.4 and 2.9. - Jun Yan, Apr 13 2024

Sheng-Liang Yang and Mei-yang Jiang, The m-Schröder paths and m-Schröder numbers, Disc. Math. (2021) Vol. 344, Issue 2, 112209. doi:10.1016/j.disc.2020.112209. See Table 1.

J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014-2020. See Fig. 23.
Jun Yan, Results on pattern avoidance in parking functions, arXiv preprint arXiv:2404.07958 [math.CO], 2024. See Theorem 4.4.

Appears to equal A260332(n)/2 for n > 0. - N. J. A. Sloane, Mar 28 2021

The sequences listed in Yang-Jiang's Table 1 appear to be A006318, A001003, A027307, A034015, A144097, A243675, A260332, A243676. - N. J. A. Sloane, Mar 28 2021

a := proc(n) option remember; if n <= 1 then return 1 fi;
-(a(n-2)*(-5440*n^7 + 42080*n^6 - 131548*n^5 + 212750*n^4 - 189160*n^3 + 90725*n^2 - 21387*n + 1890)+ a(n-1)*(-118660*n^7 + 739880*n^6 - 1876702*n^5 + 2492120*n^4 - 1855960*n^3 + 768230*n^2 - 161913*n + 13230)) / (5440*n^7 - 25760*n^6 + 43468*n^5 - 29510*n^4 + 4750*n^3 + 1945*n^2 - 468*n) end:
seq(a(n), n = 0..20);  # Peter Luschny, Apr 13 2024

a[n_] := Hypergeometric2F1[1 - n, -4 n, 2, 2];
Table[a[n], {n, 0, 20}]  (* Peter Luschny, Apr 13 2024 *)

a(n+1) = Sum_{i=0..n} Sum_{j=0..i} (-2)^(n-i)*binomial(i,j)*binomial(4*i+j, n)*binomial(n+1,i)/(n+1) (conjectured). - Michael D. Weiner, May 25 2017
a(n) = Sum_{i=1..n} binomial(4*n, i-1)*binomial(n, i)*2^(i-1)/n (conjectured). - Michael D. Weiner, Jul 24 2019 [This is correct for n>0 - Jun Yan, Apr 13 2024]
Let D(n) be the set of 4-Dyck paths with n up-steps of size 4, 4n down-steps of size 1 and never go below the x-axis. For every d in D(n), let peak(d) be the number of peaks in d. Then a(n) = Sum_{d in D(n)}2^(peak(d) - 1). - Jun Yan, Apr 13 2024
a(n) = hypergeom([1 - n, -4*n], [2], 2). - Peter Luschny, Apr 13 2024

Added a(0)=1. - N. J. A. Sloane, Mar 28 2021

More terms from Jun Yan, Apr 13 2024

A235347 Series reversion of x(1-3x^2)/(1-x^2) in odd-order powers.

Original entry on oeis.org

1, 2, 14, 130, 1382, 15906, 192894, 2427522, 31405430, 415086658, 5580629870, 76080887042, 1049295082630, 14613980359010, 205246677882078, 2903566870820610, 41337029956899222, 591796707042765954, 8514525059135909070, 123048063153362454402

Offset: 0

Fung Lam, Jan 10 2014

This sequence is implied in the solutions of magnetohydrodynamics equations in R^3 for incompressible, electrically-conducting fluids in the presence of a strong Lorentz force. a(n) = numbers of allowable magneto-vortical eddies in terms of initial conditions.

Georg Fischer, Table of n, a(n) for n = 0..1000 (first 940 terms from _Fung Lam_)
Fung Lam, On the Well-posedness of Magnetohydrodynamics Equations for Incompressible Electrically-Conducting Fluids, arXiv:1401.2029 [physics.flu-dyn], 2014-2023.

Cf. A027307, A107841, A235352 (same except for signs).

Order := 60 ;
solve(series(x*(1-3*x^2)/(1-x^2),x)=y,x) ;
convert(%,polynom) ;
seq(coeff(%,y,2*i+1),i=0..Order/2) ; # R. J. Mathar, Jul 20 2023

Table[(CoefficientList[InverseSeries[Series[x*(1-3*x^2)/(1-x^2),{x,0,40}],x],x])[[n]],{n,2,40,2}] (* Vaclav Kotesovec, Jan 29 2014 *)

v=Vec( serreverse(x*(1-3*x^2)/(1-x^2) +O(x^66) ) ); vector(#v\2,j,v[2*j-1]) \\ Joerg Arndt, Jan 14 2014

G.f.: (exp(4*Pi*i/3)*u + exp(2*Pi*i/3)*v + x/9)/x, where i=sqrt(-1),
u = (1/9)*(x^3 - 108 *x + 9*sqrt(-9 + 141*x^2 - 3*x^4))^(1/3), and
v = (1/9)*(x^3 - 108 *x - 9*sqrt(-9 + 141*x^2 - 3*x^4))^(1/3).
a(n) = [x^n] 2*Sum_{j = 1..n} ((Sum_{k = 1..n} a(k)*x^(2*k-1))^(2*j+1)), a(1) = 1, with offset by 1.
D-finite with recurrence 12*n*(2*n+1)*a(n) +(-382*n^2+391*n-90)*a(n-1) +3*(34*n^2-132*n+125)*a(n-2) -(2*n-5)*(n-3)*a(n-3)=0. - R. J. Mathar, Mar 24 2023
From Seiichi Manyama, Aug 09 2023: (Start)
a(n) = (-1)^n * Sum_{k=0..n} (-3)^k * binomial(n,k) * binomial(2*n+k+1,n) / (2*n+k+1).
a(n) = (1/n) * Sum_{k=0..n-1} 2^(n-k) * binomial(n,k) * binomial(3*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} 2^k * 3^(n-k) * binomial(n,k) * binomial(2*n,k-1) for n > 0. (End)
From Peter Bala, Sep 08 2024: (Start)
a(n) = 2*Jacobi_P(n-1, 1, n+1, 5)/n for n >= 1.
Second-order recurrence: 3*n*(2*n + 1)*(13*n - 17)*a(n) = (1222*n^3 - 2820*n^2 + 1877*n - 360)*a(n-1) - (n - 2)*(13*n - 4)*(2*n - 3)*a(n-2) with a(0) = 1 and a(1) = 2. (End)

cp's OEIS Frontend

A108447 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 no peaks of the form ud.

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A260332 Labelings of n diamond-shaped posets with 4 vertices per diamond where the labels follow the poset relations whose associated reading permutation avoids 231 in the classical sense.

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A333090 a(n) is equal to the n-th order Taylor polynomial (centered at 0) of S(x)^n evaluated at x = 1, where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the o.g.f. of the Schröder numbers A006318.

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A026000 a(n) = T(2n, n), where T is the Delannoy triangle (A008288).

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A363311 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x)^3 + A(x)^5).

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A085403 Expansion of (1-x+sqrt(1-6x+x^2))/2 in powers of x.

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A104978 Triangle read by rows, where the g.f. satisfies A(x, y) = 1 + x*A(x, y)^2 + x*y*A(x, y)^3.

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A333090 a(n) is equal to the n-th order Taylor polynomial (centered at 0) of S(x)^n evaluated at x = 1, where S(x) = (1 - x - sqrt(1 - 6x + x^2))/(2x) is the o.g.f. of the Schröder numbers A006318.

A104978 Triangle read by rows, where the g.f. satisfies A(x, y) = 1 + xA(x, y)^2 + xy*A(x, y)^3.

A235347 Series reversion of x(1-3x^2)/(1-x^2) in odd-order powers.