A104978 -id:A104978 - cp's OEIS frontend

A104979 Semidiagonal sums of triangle A104978: a(n) = Sum_{k=0..[n/2]} A104978(n-k,n-2*k).

1, 1, 4, 17, 85, 459, 2614, 15454, 93947, 583568, 3687761, 23633072, 153227250, 1003281314, 6624658716, 44062205158, 294938814921, 1985330061570, 13430612284606, 91262392343333, 622624395706714, 4263163419492661

Offset: 0

Paul D. Hanna, Mar 30 2005

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A104978.

A104979:= func< n | (&+[Binomial(3*n-4*k, 3*n-5*k)*Binomial(3*n-5*k, n-2*k)/(2*n-3*k+1): k in [0..Floor(n/2)]]) >;
[A104979(n): n in [0..30]]; // G. C. Greubel, Jun 08 2021

Table[Sum[Binomial[3*n-4*k, 3*n-5*k]*Binomial[3*n-5*k, n-2*k]/(2*n-3*k+1), {k,0,n/2}], {n,0,30}] (* G. C. Greubel, Jun 08 2021 *)

{a(n)=sum(k=0,n\2, binomial(3*n-4*k,3*n-5*k)*binomial(3*n-5*k,n-2*k)/(2*n-3*k+1))}

def A104979(n): return sum( binomial(3*n-4*k, 3*n-5*k)*binomial(3*n-5*k, n-2*k)/(2*n-3*k+1) for k in (0..n//2) )
[A104979(n) for n in (0..30)] # G. C. Greubel, Jun 08 2021

a(n) = Sum_{k=0..floor(n/2)} A104978(n-k, n-2*k).
a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-4*k, 3*n-5*k)*binomial(3*n-5*k, n-2*k)/(2*n-3*k+1).
G.f. satisfies: A(x) = 1 + x*A(x)^3 + x^2*A(x)^2. - Paul D. Hanna, May 27 2010

A383450 2nd diagonal (from right) in A104978.

Original entry on oeis.org

2, 21, 180, 1430, 10920, 81396, 596904, 4326300, 31081050, 221760825, 1573537680, 11114897976, 78215948720, 548652722520, 3838040704080, 26784871943928, 186537501038070, 1296717366119175, 8999440181955300, 62366467037593950, 431633967218324640, 2983755440056831440, 20603495011611002400, 142131208489591604400

Offset: 0

N. J. A. Sloane, May 02 2025

nonn

N. J. Wildberger and Dean Rubine, A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, Amer. Math. Monthly (2025). See table on page 12.

Cf. A104987.

From Peter Luschny, May 04 2025: (Start)
a := n -> (3*n + 4)!/(n!*2*(2 + 2*n + 1)!): seq(a(n), n = 0..23);
gf := 2*hypergeom([5/3, 7/3], [5/2], (27*x)/4):
ser := series(gf, x, 25): seq(coeff(ser, x, k), k = 0..23);  (End)

From Peter Luschny, May 04 2025: (Start)
a(n) = (3*n + 4)! / (2*n!*(3 + 2*n)!).
a(n) = [x^n] 2*hypergeom([5/3, 7/3], [5/2], (27*x)/4).  (End)

A383451 3nd diagonal (from right) in A104978.

Original entry on oeis.org

5, 84, 990, 10010, 92820, 813960, 6864396, 56241900, 450675225, 3548173200, 27536909400, 211183061544, 1603426948760, 12070359895440, 90193956545880, 669621798598200, 4943243777508855, 36308086251336900, 265483485367681350, 1933360478165412450, 14028103934595550800, 101447684961932268960, 731424072912190585200, 5258854714114889362800

Offset: 0

N. J. A. Sloane, May 02 2025

nonn

N. J. Wildberger and Dean Rubine, A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, Amer. Math. Monthly (2025). See table on page 12.

Cf. A104987.

From Peter Luschny, May 04 2025:  (Start)
a := n -> (3*n + 6)!/(6*n!*(3 + 2*n + 1)!): seq(a(n), n = 0..23);
gf := 5*hypergeom([7/3, 8/3], [5/2], (27*x)/4):
ser := series(gf, x, 25): seq(coeff(ser, x, k), k = 0..23);  (End)

From Peter Luschny, May 04 2025: (Start)
a(n) = (3*n + 6)!/(6*n!*(3 + 2*n + 1)!).
a(n) = [x^n] 5*hypergeom([7/3, 8/3], [5/2], (27*x)/4). (End)

A383452 Column 3 in A104978.

Original entry on oeis.org

0, 0, 0, 12, 165, 1430, 10010, 61880, 352716, 1899240, 9806280, 49031400, 239028075, 1141710570, 5362579950, 24837212400, 113678010600, 515030986800, 2312957340720, 10307744670600, 45626928615450, 200758485907980, 878623171119540, 3826892034209552, 16596215454480200, 71691488703052400, 308585103547921200, 1323929637802371600

Offset: 0

N. J. A. Sloane, May 02 2025

nonn

N. J. Wildberger and Dean Rubine, A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, Amer. Math. Monthly (2025). See table on page 12.

Cf. A104987.

a := n -> ifelse(n < 3, 0, (3 + 2*n)! / (6*(n - 3)!*(n + 4)!)): seq(a(n), n = 0..27);
y := sqrt(1 - 4*x): gf := (1/(2*(x*y)^4))*((210*x^4 - 420*x^3 + 252*x^2 - 60*x + 5)/y -(32*x^4 - 176*x^3 + 162*x^2 - 50*x + 5)): ser := series(gf, x, 34):
seq(coeff(ser, x, n), n = 0..27); # Peter Luschny, May 04 2025

From Peter Luschny, May 04 2025: (Start)
a(n) = (3 + 2*n)! / (6*(n - 3)!*(n + 4)!) for n >= 3.
a(n) = [x^n] (1/(2*(x*y)^4))*((210*x^4 - 420*x^3 + 252*x^2 - 60*x + 5)/y -(32*x^4 - 176*x^3 + 162*x^2 - 50*x + 5)) where y = sqrt(1 - 4*x). (End)

A027307 Number of paths from (0,0) to (3n,0) that stay in first quadrant (but may touch horizontal axis) and where each step is (2,1), (1,2) or (1,-1).

Original entry on oeis.org

1, 2, 10, 66, 498, 4066, 34970, 312066, 2862562, 26824386, 255680170, 2471150402, 24161357010, 238552980386, 2375085745978, 23818652359682, 240382621607874, 2439561132029314, 24881261270812490, 254892699352950850

Offset: 0

Emeric Deutsch

These are the 3-Schroeder numbers according to Yang-Jiang (2021). - N. J. A. Sloane, Mar 28 2021
Equals row sums of triangle A104978 which has g.f. F(x,y) that satisfies: F = 1 + x*F^2 + x*y*F^3. - Paul D. Hanna, Mar 30 2005
a(n) counts ordered complete ternary trees with 2*n + 1 leaves, where the internal vertices come in two colors and such that each vertex and its rightmost child have different colors. See [Drake, Example 1.6.9]. An example is given below. - Peter Bala, Sep 29 2011
a(n) for n >= 1 is the number of compact coalescent histories for matching lodgepole gene trees and species trees with n cherries and 2n+1 leaves. - Noah A Rosenberg, Jun 21 2022
a(n) is the maximum number of distinct sets that can be obtained as complete parenthesizations of “S_1 union S_2 intersect S_3 union S_4 intersect S_5 union ... union S_{2*n} intersect S_{2*n+1}”, where n union and n intersection operations alternate, starting with a union, and S_1, S_2, ... , S_{2*n+1} are sets. - Alexander Burstein, Nov 22 2023

			a(2) = 10. Internal vertices colored either b(lack) or w(hite); 5 uncolored leaf vertices shown as o.
........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..
....................................................
........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....
...............................
From _Alexander Burstein_, Feb 14 2025: (Start)
a(2) = 10 as the maximum number of distinct sets obtained as complete parenthesizations of S_1 u(nion) S_2 (i)n(tersect) S_3 u(nion) S_4 (i)n(tersect) S_5:
S_1 u (S_2 n (S_3 u (S_4 n S_5))),
S_1 u (S_2 n ((S_3 u S_4) n S_5)) = S_1 u ((S_2 n (S_3 u S_4)) n S_5),
S_1 u ((S_2 n S_3) u (S_4 n S_5)) = (S_1 u (S_2 n S_3)) u (S_4 n S_5),
S_1 u (((S_2 n S_3) u S_4) n S_5),
(S_1 u S_2) n (S_3 u (S_4 n S_5)),
(S_1 u S_2) n ((S_3 u S_4) n S_5) = ((S_1 u S_2) n (S_3 u S_4)) n S_5,
((S_1 u S_2) n S_3) u (S_4 n S_5),
(S_1 u (S_2 n (S_3 u S_4))) n S_5,
(S_1 u ((S_2 n S_3) u S_4)) n S_5 = ((S_1 u (S_2 n S_3)) u S_4) n S_5,
(((S_1 u S_2) n S_3) u S_4) n S_5. (End)

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.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
Gi-Sang Cheon, S.-T. Jin and L. W. Shapiro, A combinatorial equivalence relation for formal power series, Linear Algebra and its Applications, Available online 30 March 2015.
Emeric Deutsch, Problem 10658, American Math. Monthly, 107, 2000, 368-370.
F. Disanto and N. A. Rosenberg, Enumeration of compact coalescent histories for matching gene trees and species trees, J. Math. Biol 78 (2019), 155-188.
B. Drake, An inversion theorem for labeled trees and some limits of areas under lattice paths (Example 1.6.9), A dissertation presented to the Faculty of the Graduate School of Arts and Sciences of Brandeis University.
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Zhen-ji Tian and Shu-ping Dou, Enumerations of 3 di-sk trees, J. Lanzhou Univ. Tech. (2024) Vol. 50, No. 6, 144-149. See p. 146.
J. Winter, M. M. Bonsangue and J. J. M. M. Rutten, Context-free coalgebras, 2013.
Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 11.
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)
Anssi Yli-Jyrä and Carlos Gómez-Rodríguez, Generic Axiomatization of Families of Noncrossing Graphs in Dependency Parsing, arXiv:1706.03357 [cs.CL], 2017.
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.

Cf. A104978. A196201.
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
Apart from first term, this is 2*A034015. - N. J. A. Sloane, Mar 28 2021

a[n_] := ((n+1)*(2n)!*Hypergeometric2F1[-n, 2n+1, n+2, -1]) / (n+1)!^2;
Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Nov 14 2011, after Pari *)
a[n_] := If[n == 0, 1, 2*Hypergeometric2F1[1 - n, -2 n, 2, 2]];
Table[a[n], {n, 0, 19}]  (* Peter Luschny, Nov 08 2021 *)

a(n)=if(n<1,n==0,sum(i=0,n-1,2^(i+1)*binomial(2*n,i)*binomial(n,i+1))/n)

a(n)=sum(k=0,n,binomial(2*n+k,n+2*k)*binomial(n+2*k,k)/(n+k+1)) \\ Paul D. Hanna

a(n)=sum(k=0,n, binomial(n,k)*binomial(2*n+k+1,n)/(2*n+k+1) ) /* Michael Somos, May 23 2005 */

G.f.: (2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3.
a(n) = (1/n) * Sum_{i=0..n-1} 2^(i+1)*binomial(2*n, i)*binomial(n, i+1), n>0.
a(n) = 2*A034015(n-1), n>0.
a(n) = Sum_{k=0..n} C(2*n+k, n+2*k)*C(n+2*k, k)/(n+k+1). - Paul D. Hanna, Mar 30 2005
Given g.f. A(x), y=A(x)x satisfies 0=f(x, y) where f(x, y)=x(x-y)+(x+y)y^2 . - Michael Somos, May 23 2005
Series reversion of x(Sum_{k>=0} a(k)x^k) is x(Sum_{k>=0} A085403(k)x^k).
G.f. A(x) satisfies A(x)=A006318(x*A(x)). - Vladimir Kruchinin, Apr 18 2011
The function B(x) = x*A(x^2) satisfies B(x) = x+x*B(x)^2+B(x)^3 and hence B(x) = compositional inverse of x*(1-x^2)/(1+x^2) = x+2*x^3+10*x^5+66*x^7+.... Let f(x) = (1+x^2)^2/(1-4*x^2+x^4) and let D be the operator f(x)*d/dx. Then a(n) equals 1/(2*n+1)!*D^(2*n)(f(x)) evaluated at x = 0. For a refinement of this sequence see A196201. - Peter Bala, Sep 29 2011
D-finite with recurrence: 2*n*(2*n+1)*a(n) = (46*n^2-49*n+12)*a(n-1) - 3*(6*n^2-26*n+27)*a(n-2) - (n-3)*(2*n-5)*a(n-3). - Vaclav Kotesovec, Oct 08 2012
a(n) ~ sqrt(50+30*sqrt(5))*((11+5*sqrt(5))/2)^n/(20*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 08 2012. Equivalently, a(n) ~ phi^(5*n + 1) / (2 * 5^(1/4) * sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 07 2021
a(n) = 2*hypergeom([1 - n, -2*n], [2], 2) for n >= 1. - Peter Luschny, Nov 08 2021
From Peter Bala, Jun 16 2023: (Start)
P-recursive: n*(2*n + 1)*(5*n - 7)*a(n) = (110*n^3 - 264*n^2 + 181*n - 36)*a(n-1) + (n - 2)*(2*n - 3)*(5*n - 2)*a(n-2) with a(0) = 1 and a(1) = 2.
The g.f. A(x) = 1 + 2*x + 10*x^2 + 66*x^3 + ... satisfies A(x)^2 = (1/x) * the series reversion of x*((1 - x)/(1 + x))^2.
Define b(n) = [x^(2*n)] ( (1 + x)/(1 - x) )^n = (1/2) * [x^n] ((1 + x)/(1 - x))^(2*n) = A103885(n). Then A(x) = exp( Sum_{n >= 1} b(n)*x^n/n ). (End)
a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * 2^(n-k) * binomial(n,k) * binomial(3*n-k,n-1-k) for n > 0. - Seiichi Manyama, Aug 09 2023

A001002 Number of dissections of a convex (n+2)-gon into triangles and quadrilaterals by nonintersecting diagonals.

Original entry on oeis.org

1, 1, 3, 10, 38, 154, 654, 2871, 12925, 59345, 276835, 1308320, 6250832, 30142360, 146510216, 717061938, 3530808798, 17478955570, 86941210950, 434299921440, 2177832612120, 10959042823020, 55322023332420, 280080119609550, 1421744205767418, 7234759677699954

Offset: 0

N. J. A. Sloane

a(n+1) is number of (2,3)-rooted trees on n nodes.
This sequence appears to be a transform of the Fibonacci numbers A000045. This sequence is to the Fibonacci numbers as the Catalan numbers A000108 is to the all ones sequence. See link to Mathematica program. - Mats Granvik, Dec 30 2017
a(n) is the number of parking functions of size n avoiding the patterns 231, 312, and 321. - Lara Pudwell, Apr 10 2023

			a(3)=10 because a convex pentagon can be dissected in 5 ways into triangles (draw 2 diagonals from any of the 5 vertices) and in 5 ways into a triangle and a quadrilateral (draw any of the 5 diagonals).

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 211 (3.2.73-74)
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1372 (first 101 terms from T. D. Noe)
Ayomikun Adeniran and Lara Pudwell, Pattern avoidance in parking functions, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
Paul Barry, On the Central Coefficients of Bell Matrices, J. Int. Seq. 14 (2011) # 11.4.3, example 7.
D. Birmajer, J. B. Gil, and M. D. Weiner, Colored partitions of a convex polygon by noncrossing diagonals, arXiv preprint arXiv:1503.05242 [math.CO], 2015.
I. M. H. Etherington, Non-associate powers and a functional equation, Math. Gaz. 21 (1937), 36-39; addendum 21 (1937), 153.
I. M. H. Etherington, On non-associative combinations, Proc. Royal Soc. Edinburgh, 59 (Part 2, 1938-39), 153-162.
I. M. H. Etherington, Some problems of non-associative combinations, Edinburgh Math. Notes, 32 (1940), 1-6.
I. M. H. Etherington, Some problems of non-associative combinations (I), Edinburgh Math. Notes, 32 (1940), pp. i-vi. [Annotated scanned copy].
A. Erdelyi and I. M. H. Etherington, Some problems of non-associative combinations (II), Edinburgh Math. Notes, 32 (1940), pp. vii-xiv.
Mats Granvik, Mathematica program to compute the sequence as a transform of the Fibonacci numbers.
Nancy S. S. Gu, Nelson Y. Li, and Toufik Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 395
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
T. Motzkin, The hypersurface cross ratio, Bull. Amer. Math. Soc., 51 (1945), 976-984.
T. S. Motzkin, Relations between hypersurface cross ratios and a combinatorial formula for partitions of a polygon, for permanent preponderance and for non-associative products, Bull. Amer. Math. Soc., 54 (1948), 352-360.
Thomas M. Richardson, The three 'R's and Dual Riordan Arrays, arXiv:1609.01193 [math.CO], 2016.
A. Schuetz and G. Whieldon, Polygonal Dissections and Reversions of Series, arXiv preprint arXiv:1401.7194 [math.CO], 2014.
Index entries for reversions of series

n*a(n) = A038112(n-1), n > 0.

Cf. A104978, A181997, A181998, A209441, A209442, A213684 (log).

List([0..25], n->Sum([0..Int(n/2)],k->Binomial(2*n-k,n+k)*Binomial(n+k,k)/(n+1))); # Muniru A Asiru, Mar 30 2018

a:= proc(n) option remember; `if`(n<2, 1, (n*(22*n-11)*
      a(n-1) + (9*n-6)*(3*n-4)*a(n-2))/(5*n*(n+1)))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 21 2021

Rest[CoefficientList[InverseSeries[Series[y - y^2 - y^3, {y, 0, 30}], x], x]]
a[n_] := CatalanNumber[n]*Hypergeometric2F1[1/2-n/2, -n/2, -2n, -4]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 20 2015, after Peter Luschny *)
a[n_] := a[n] = If[n == 0, 1, Sum[a[i] a[n - 1 - i], {i, 0, n - 1}] + Sum[a[i] a[j] a[n - 2 - i - j], {i, 0, n - 2}, {j, 0, n - 2 - i}]];
Table[a[n], {n, 0, 30}] (* Li Han, Jan 02 2021 *)

T(n,k):=if n<0 or k<0 then 0 else if nVladimir Kruchinin, Oct 03 2014 */

a(n)=if(n<0,0,polcoeff(serreverse(x-x^2-x^3+x^2*O(x^n)),n+1))

a(n)=if(n<0,0,sum(k=0,n\2,(2*n-k)!/k!/(n-2*k)!)/(n+1)!)

a(n)=sum(k=0,n\2,binomial(2*n-k,n+k)*binomial(n+k,k))/(n+1) \\ Hanna

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=1); A=1+(1/x)*sum(m=1, n+1, Dx(m-1, (x^2+x^3 +x^2*O(x^n))^m/m!)); polcoeff(A, n)}  \\ Paul D. Hanna, Jun 22 2012

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=1); A=exp(sum(m=1, n+1, Dx(m-1, (x^2+x^3 +x^2*O(x^n))^m/x/m!)+x*O(x^n))); polcoeff(A, n)}  \\ Paul D. Hanna, Jun 22 2012

A001002 = lambda n: catalan_number(n)*hypergeometric([1/2-n/2, -n/2], [-2*n], -4) if n>0 else 1
[A001002(n).n(100).round() for n in range(24)] # Peter Luschny, Oct 03 2014

G.f. (offset 1) is series reversion of x - x^2 - x^3.
a(n) = (1/(n+1))*Sum_{k=ceiling(n/2)..n} binomial(n+k, k)*binomial(k, n-k). - Len Smiley
D-finite with recurrence 5*n*(n+1) * a(n) = 11*n*(2*n-1) * a(n-1) + 3*(3*n-2)*(3*n-4) * a(n-2). - Len Smiley
G.f.: (4*sin(asin((27*x+11)/16)/3)-1)/(3*x). - Paul Barry, Feb 02 2005
G.f. satisfies: A(x) = 1 + x*A(x)^2 + x^2*A(x)^3. - Paul D. Hanna, Jun 22 2012
Antidiagonal sums of triangle A104978 which has g.f. F(x,y) that satisfies: F = 1 + x*F^2 + x*y*F^3. - Paul D. Hanna, Mar 30 2005
a(n) = Sum_{k=0..floor(n/2)} C(2*n-k, n+k)*C(n+k, k)/(n+1). - Paul D. Hanna, Mar 30 2005
G.f. satisfies: x = Sum_{n>=1} 1/(1+x*A(x))^(2*n) * Product_{k=1..n} (1 - 1/(1+x*A(x))^k). - Paul D. Hanna, Apr 05 2012
G.f.: 1 + (1/x)*Sum_{n>=1} d^(n-1)/dx^(n-1) (x^2+x^3)^n/n!. - Paul D. Hanna, Jun 22 2012
G.f.: exp( Sum_{n>=1} d^(n-1)/dx^(n-1) ((x^2+x^3)^n/x)/n! ). - Paul D. Hanna, Jun 22 2012
Logarithmic derivative yields A213684. - Paul D. Hanna, Jun 22 2012
a(n) ~ 3^(3*n+3/2) / (2 * sqrt(2*Pi) * 5^(n+1/2) * n^(3/2)). - Vaclav Kotesovec, Mar 09 2014
a(n) = Catalan(n)*hypergeom([1/2-n/2, -n/2], [-2*n], -4) for n>0. - Peter Luschny, Oct 03 2014
a(n) = [x^n] 1/(1 - x - x^2)^(n+1)/(n + 1). - Ilya Gutkovskiy, Mar 29 2018
a(n) = -Sum_{i=1..n} A217596(i) * a(n-i) for n>0. - Muhammed Sefa Saydam, Jan 27 2025
a(n) = -Sum_{i=1..n+2} A217596(i) * A217596(n-i+2) for n >= 0. - Muhammed Sefa Saydam, Jul 24 2025

Revised by Emeric Deutsch and Len Smiley, Jun 05 2005

A383439 a(n) = (5n)!/((n!)^2(3*n + 1)!).

Original entry on oeis.org

1, 5, 180, 10010, 678300, 51482970, 4206302100, 361913666400, 32356261929420, 2979510862285100, 280884023785324960, 26990111025198348300, 2634899457411931245900, 260690108634780944767200, 26088052554768282442056000, 2636591265602354831196585600, 268771779551047800167424355500

Offset: 0

Peter Luschny, May 03 2025

nonn

N. J. Wildberger and Dean Rubine, A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, Amer. Math. Monthly (2025), p. 12.

Cf. A104978, A383440.

Maple
```
a := n -> (5*n)!/((n!)^2*(3*n + 1)!):
```

Array[(5*#)!/((#!)^2*(3*# + 1)!) &, 17, 0] (* Michael De Vlieger, May 03 2025 *)

a(n) = A104978(2*n, n), main diagonal of the Bi-Tri array C[m_2, m_3] in the terminology of Wildberger-Rubine.

a(n) ~ 4*(20*n - 3)*3^(5/2-3*n)*5^(5*n-1/2)/(n*(25 + 936*n + 2592*n^2)*Pi). - Stefano Spezia, May 03 2025

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).

cp's OEIS Frontend

A104979 Semidiagonal sums of triangle A104978: a(n) = Sum_{k=0..[n/2]} A104978(n-k,n-2*k).

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A383450 2nd diagonal (from right) in A104978.

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A383451 3nd diagonal (from right) in A104978.

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A383452 Column 3 in A104978.

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A027307 Number of paths from (0,0) to (3n,0) that stay in first quadrant (but may touch horizontal axis) and where each step is (2,1), (1,2) or (1,-1).

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A001002 Number of dissections of a convex (n+2)-gon into triangles and quadrilaterals by nonintersecting diagonals.

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A383439 a(n) = (5*n)!/((n!)^2*(3*n + 1)!).

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A383439 a(n) = (5n)!/((n!)^2(3*n + 1)!).