A006013 -id:A006013 - cp's OEIS frontend

A117671 a(n) = binomial(3*n+1, n+1).

1, 6, 35, 210, 1287, 8008, 50388, 319770, 2042975, 13123110, 84672315, 548354040, 3562467300, 23206929840, 151532656696, 991493848554, 6499270398159, 42671977361650, 280576272201225, 1847253511032930, 12176310231149295, 80347448443237920, 530707489338171600

Offset: 0

Zerinvary Lajos, Apr 12 2006

			if n=0 then C(3*0+1,0+1) = C(1,1) = 1.
if n=10 then C(3*10+1,10+1) = C(31,11) = 84672315.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Milan Janjić, Two Enumerative Functions.

Cf. A025174 (binomial(3n-1,n-1)), A006013.

Cf. A258993, A001764, A004319, A005809, A013698, A045721, A165817, A236194.

a117671 n = a258993 (2 * n + 1) n  -- Reinhard Zumkeller, Jun 22 2015

seq(binomial(3*n+1,n+1),n=0..30); # Robert Israel, Oct 10 2017

Table[Binomial[3n+1,n+1],{n,0,20}] (* Harvey P. Dale, Jul 19 2011 *)

vector(30, n, n--; binomial(3*n+1, n+1)) \\ Altug Alkan, Nov 04 2015

G.f.: (2*(-1+Hypergeometric2F1[-(1/3),1/3,-(1/2),(27*x)/4]))/(3*x). - Harvey P. Dale, Jul 19 2011
G.f.: A(x) = B'(x)/B(x)-B'(x)-1/x, where B(x) = 4/3*sin(1/3*asin(sqrt((27*x)/4)))^2. - Vladimir Kruchinin, Nov 26 2014
a(n) = A258993(2*n+1, n). - Reinhard Zumkeller, Jun 22 2015
From Peter Bala, Nov 04 2015: (Start)
With an extra initial term equal to 1, the o.g.f. equals f(x)/g(x)^2, where f(x) is the o.g.f. for A005809 and g(x) is the o.g.f. for A001764.
More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(3*n + k,n). Cf. A045721 (k = 1), A025174 (k = 2), A004319 (k = 3), A236194 (k = 4), A013698 (k = 5), A165817 (k = -1). (End)
a(n) = [x^(2*n)] 1/(1 - x)^(n+2). - Ilya Gutkovskiy, Oct 10 2017
a(n+1) = 3*(3*n+2)*(3*n+4)*a(n)/(2*(n+2)*(2*n+1)). - Robert Israel, Oct 10 2017
a(n) ~ 3^(3*n+3/2) / (2^(2*n+1) * sqrt(Pi*n)). - Amiram Eldar, Sep 05 2025

A251573 E.g.f.: exp(3xG(x)^2) / G(x)^2 where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 3, 21, 261, 4833, 120303, 3778029, 143531433, 6404711553, 328447585179, 19037277446949, 1230842669484717, 87829738967634849, 6856701559496841159, 581343578623728854397, 53196439113856500195537, 5225543459274294130169601, 548468830470032135590262067, 61258398893626609968686844597

Offset: 0

Paul D. Hanna, Dec 06 2014

nonn

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 21*x^3/3! + 261*x^4/4! + 4833*x^5/5! +...
such that A(x) = exp(3*x*G(x)^2) / G(x)^2
where G(x) = 1 + x*G(x)^3 is the g.f. of A001764:
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
The e.g.f. satisfies:
A(x) = 1 + x/A(x) + 5*x^2/(2!*A(x)^2) + 54*x^3/(3!*A(x)^3) + 945*x^4/(4!*A(x)^4) + 23328*x^5/(5!*A(x)^5) + 750141*x^6/(6!*A(x)^6) + 29859840*x^7/(7!*A(x)^7) +...+ 3^(n-1)*(n+1)^(n-3)*(n+3) * x^n/(n!*A(x)^n) +...
Note that
A'(x) = exp(3*x*G(x)^2) = 1 + 3*x + 21*x^2/2! + 261*x^3/3! + 4833*x^4/4! +...
LOGARITHMIC DERIVATIVE.
The logarithm of the e.g.f. begins:
log(A(x)) = x + 2*x^2/2 + 7*x^3/3 + 30*x^4/4 + 143*x^5/5 +...
and so A'(x)/A(x) = G(x)^2.
TABLE OF POWERS OF E.G.F.
Form a table of coefficients of x^k/k! in A(x)^n as follows.
n=1: [1, 1,  3,   21,   261,   4833,  120303,   3778029, ...];
n=2: [1, 2,  8,   60,   744,  13536,  330912,  10232928, ...];
n=3: [1, 3, 15,  123,  1557,  28179,  680427,  20771235, ...];
n=4: [1, 4, 24,  216,  2832,  51552, 1237248,  37404288, ...];
n=5: [1, 5, 35,  345,  4725,  87285, 2094975,  62949825, ...];
n=6: [1, 6, 48,  516,  7416, 139968, 3378528, 101278944, ...];
n=7: [1, 7, 63,  735, 11109, 215271, 5250987, 157613463, ...];
n=8: [1, 8, 80, 1008, 16032, 320064, 7921152, 238878720, ...]; ...
in which the main diagonal begins (see A251583):
[1, 2, 15, 216, 4725, 139968, 5250987, 238878720, ...]
and is given by the formula:
[x^n/n!] A(x)^(n+1) = 3^(n-1) * (n+1)^(n-2) * (n+3) for n>=0.

Paul D. Hanna, Table of n, a(n) for n = 0..200

Cf. A251583, A251663, A001764, A006013.

Cf. Variants: A243953, A251574, A251575, A251576, A251577, A251578, A251579, A251580.

Flatten[{1,1,Table[Sum[3^k * n!/k! * Binomial[3*n-k-3, n-k] * (k-1)/(n-1),{k,0,n}],{n,2,20}]}] (* Vaclav Kotesovec, Dec 07 2014 *)
Flatten[{1,1,RecurrenceTable[{27*(n-2)*a[n-2]-3*(3*n-8)*(15-13*n+3*n^2)*a[n-1]+2*(n-3)*(2*n-3)*a[n]==0,a[2]==3,a[3]==21},a,{n,20}]}] (* Vaclav Kotesovec, Dec 07 2014 *)

{a(n) = local(G=1);for(i=1,n,G=1+x*G^3 +x*O(x^n)); n!*polcoeff(exp(3*x*G^2)/G^2, n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = if(n==0||n==1, 1, sum(k=0, n, 3^k * n!/k! * binomial(3*n-k-3,n-k) * (k-1)/(n-1) ))}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^3 be the g.f. of A001764, then the e.g.f. A(x) of this sequence satisfies:
(1) A'(x)/A(x) = G(x)^2.
(2) A'(x) = exp(3*x*G(x)^2).
(3) A(x) = exp( Integral G(x)^2 dx ).
(4) A(x) = exp( Sum_{n>=1} A006013(n-1)*x^n/n ), where A006013(n-1) = binomial(3*n-2,n)/(2*n-1).
(5) A(x) = F(x/A(x)) where F(x) is the e.g.f. of A251583.
(6) A(x) = Sum_{n>=0} A251583(n)*(x/A(x))^n/n! where A251583(n) = 3^(n-1) * (n+1)^(n-3) * (n+3).
(7) [x^n/n!] A(x)^(n+1) = 3^(n-1) * (n+1)^(n-2) * (n+3).
a(n) = Sum_{k=0..n} 3^k * n!/k! * binomial(3*n-k-3, n-k) * (k-1)/(n-1) for n>1.
Recurrence (for n>3): 2*(n-3)*(2*n-3)*a(n) = 3*(3*n-8)*(3*n^2 - 13*n + 15)*a(n-1) - 27*(n-2)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 3^(3*n-7/2) * n^(n-2) / (2^(2*n-5/2) * exp(n-1)). - Vaclav Kotesovec, Dec 07 2014

A118969 a(n) = 2binomial(5n+1,n)/(4*n+2).

Original entry on oeis.org

1, 2, 11, 80, 665, 5980, 56637, 556512, 5620485, 57985070, 608462470, 6474009360, 69682358811, 757366074080, 8300675584120, 91634565938880, 1018002755977245, 11372548404732930, 127677890035721025, 1439777493407492640

Offset: 0

Paul Barry, May 07 2006

A quadrisection of A118968.

If y = x + 2*x^3 + x^5, the series reversion is x = y - 2*y^3 + 11*y^5 - 80*y^7 + 665*y^9 - ... - R. J. Mathar, Sep 29 2012

			a(3) = 80 = sum of top row terms in M^n = (35 + 35 + 9 + 1).

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k, arXiv:2204.14023 [math.CO], 2022.
Karol A. Penson and Karol Zyczkowski, Product of Ginibre matrices: Fuss-Catalan and Raney distribution, arXiv:1103.3453 [math-ph], 2011.
Karol A. Penson and Karol Zyczkowski, Product of Ginibre matrices: Fuss-Catalan and Raney distribution, Phys. Rev. E 83, 061118 (2011).
Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 7.
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. A000292, A118968, A006013, A233832, A234505, A234868, A002294.

[2*Binomial(5*n+1,n)/(4*n+2): n in [0..20]]; // Vincenzo Librandi, Aug 12 2011

Table[2*Binomial[5n+1,n]/(4n+2),{n,0,20}] (* Harvey P. Dale, Aug 21 2011 *)

a(n)=2*binomial(5*n+1,n)/(4*n+2); \\ Joerg Arndt, Apr 20 2013

From Gary W. Adamson, Aug 11 2011: (Start)
a(n) is sum of top row terms in M^n, where M is an infinite square production matrix with the tetrahedral series in each column (A000292), as follows:
   1,  1,  0,  0,  0,  0, ...
   4,  1,  1,  0,  0,  0, ...
  10, 10,  4,  1,  0,  0, ...
  20, 20, 10,  4,  1,  0, ...
  35, 35, 20, 10,  4,  1, ...
  ... (End)
G.f.: hypergeom([1/5, 2/5, 3/5, 4/5],[1/2, 3/4, 5/4], 3125*x/256)^2. - Mark van Hoeij, Apr 19 2013
a(n) = 2*binomial(5n+1,n-1)/n for n>0, a(0)=1. - Bruno Berselli, Jan 19 2014
D-finite with recurrence 8*n*(4*n+1)*(2*n+1)*(4*n-1)*a(n) - 5*(5*n+1)*(5*n-3)*(5*n-2)*(5*n-1)*a(n-1) = 0. - R. J. Mathar, Oct 10 2014
G.f. A(x) satisfies: A(x) = 1 / (1 - x * A(x)^2)^2. - Ilya Gutkovskiy, Nov 13 2021

A130565 Member k=6 of a family of generalized Catalan numbers.

Original entry on oeis.org

1, 6, 57, 650, 8184, 109668, 1533939, 22137570, 327203085, 4928006512, 75357373305, 1166880131820, 18259838103852, 288308609783760, 4587430875645660, 73484989079268690, 1184104656043939071

Offset: 1

Wolfdieter Lang, Jul 13 2007

The generalized Catalan numbers C(k,n):= binomial(k*n+1,n)/(k*n+1) become for negative k=-|k|, with |k|>=2, ((-1)^(n-1))*binomial((|k|+1)*n-2,n)/(|k|*n-1), n>=0.
For the members of the family C(k,n), k=2..9, see A130564.
The family c(k,n):=binomial((k+1)*n-2,n)/(k*n-1), n>=1, has the members A006013, A006632, A118971,for k=2,3,4 respectively (but the offset there is 0) and A130564 for k=5.

Harvey P. Dale, Table of n, a(n) for n = 1..806
Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

Cf. k=5 member A130564. A006013, A006632, A118971,

Table[Binomial[7n-2,n]/(6n-1),{n,20}] (* Harvey P. Dale, Feb 25 2013 *)

a(n) = binomial((k+1)*n-2,n)/(k*n-1), with k=6.
G.f.: inverse series of y*(1-y)^6.
a(n) = (6/7)*binomial(7*n,n)/(7*n-1). [Bruno Berselli, Jan 17 2014]
From Wolfdieter Lang, Feb 06 2020: (Start)
G.f.: (6/7)*(1 - hypergeom([-1, 1, 2, 3, 4, 5]/7, [1, 2, 3, 4, 5]/6, (7^7/6^6)*x)).
E.g.f.: (6/7)*(1 - hypergeom([-1, 1, 2, 3, 4, 5]/7, [1, 2, 3, 4, 5, 6]/6, (7^7/6^6)*x)). (End)

A051255 Number of cyclically symmetric transpose complement plane partitions in a 2n X 2n X 2n box.

Original entry on oeis.org

1, 1, 2, 11, 170, 7429, 920460, 323801820, 323674802088, 919856004546820, 7434724817843114428, 170943292930264547814443, 11183057455425265737399150652, 2081853548182272792243789109645876

Offset: 0

N. J. A. Sloane

Hankel transform of A006013 without initial term is this sequence without initial term. - Michael Somos, May 15 2022

			For n=0 there is the empty partition by convention so a(0)=1. For n=1 there is a single cyclically symmetric transpose complement plane partition in a 2 X 2 X 2 box so a(1)=1.
G.f. = 1 + x + 2*x^2 + 11*x^3 + 170*x^4 + 7429*x^5 + 920460*x^6 + 323801820*x^7 + ... - _Michael Somos_, May 15 2022

D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; Eq. (6.15), p. 199 (corrected).

Vincenzo Librandi, Table of n, a(n) for n = 0..60
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Paul Barry, Chebyshev moments and Riordan involutions, arXiv:1912.11845 [math.CO], 2019.
Paul Barry, Extensions of Riordan Arrays and Their Applications, Mathematics (2025) Vol. 13, No. 2, 242. See p. 24.
M. T. Batchelor, J. de Gier and B. Nienhuis, The quantum symmetric XXZ chain at Delta=-1/2, alternating sign matrices and plane partitions, arXiv:cond-mat/0101385 [cond-mat.stat-mech], 2001. See N_8(2n).
D. M. Bressoud, Corrections: Proofs and Confirmations
N. T. Cameron, Random walks, trees and extensions of Riordan group techniques, Dissertation, Howard University, 2002.
Naiomi Cameron and J. E. McLeod, Returns and Hills on Generalized Dyck Paths, Journal of Integer Sequences, Vol. 19, 2016, #16.6.1.
J. de Gier, Loops, matchings and alternating-sign matrices, arXiv:math/0211285 [math.CO], 2002.
I. Gessel and G. Xin, The generating function of ternary trees and continued fractions, arXiv:math/0505217 [math.CO], 2005.
Anatol N. Kirillov, Notes on Schubert, Grothendieck and key polynomials, SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 034, 56 p. (2016).
Yaping Liu, On the Recursiveness of Pascal Sequences, Global J. of Pure and Appl. Math. (2022) Vol. 18, No. 1, 71-80.

Cf. A006013, A049504.

A051255 := proc(n) local i; mul((3*i+1)*(6*i)!*(2*i)!/((4*i)!*(4*i+1)!),i=0..n-1); end;

a[n_] := Product[(3*i+1)*(6*i)!*(2*i)!/((4*i)!*(4*i+1)!), {i, 0, n-1}]; Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Feb 25 2014 *)

a(n)=prod(i=0,n-1,(3*i+1)*(6*i)!*(2*i)!/((4*i)!*(4*i+1)!)); \\ Joerg Arndt, Feb 25 2014

A051255(n)=prod(i=0,n-1,(3*i+1)*binomial(6*i,2*i)/binomial(4*i+1,2*i)/(2*i+1)) \\ M. F. Hasler, Oct 04 2018

a(n) ~ exp(1/72) * GAMMA(1/3)^(2/3) * n^(7/72) * 3^(3*n^2 - 3*n/2 + 11/72) / (A^(1/6) * Pi^(1/3) * 2^(4*n^2 - n - 1/18)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Feb 28 2015

a(n) = Product_{i=0..n-1} (3i+1) C(6i,2i)/(C(4i+1,2i)*(2i+1)), using [Bressoud, Corrections, p. 199: N8]. - M. F. Hasler, Oct 04 2018

More terms from Michel ten Voorde

Missing a(0)=1 term added by Michael Somos, Feb 25 2014

A236194 a(n) = binomial(3n+1, n-1).

Original entry on oeis.org

1, 7, 45, 286, 1820, 11628, 74613, 480700, 3108105, 20160075, 131128140, 854992152, 5586853480, 36576848168, 239877544005, 1575580702584, 10363194502115, 68248282427325, 449972009097765, 2969831763694950, 19619725782651120, 129728497393775280, 858478958817125100

Offset: 1

Bruno Berselli, Jan 20 2014

This sequence is related to A006013 by a(n)/n = A006013(n)/2.

Bruno Berselli, Table of n, a(n) for n = 1..100
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.

Cf. A006013; A025174: C(3n-1, n-1); A117671: C(3n+1, n+1).
Second column of the triangle A159841.
Third column of the triangle A119301.
Cf. A001764, A004319, A005809, A013698, A045721, A165817.
Cf. A045721, A117671.

Magma
```
[Binomial(3*n+1,n-1): n in [1..30]];
```
Mathematica
```
Table[Binomial[3n+1, n-1], {n, 30}]
```

makelist(binomial(3*n+4,n),n,0,40); /* Emanuele Munarini, Oct 14 2014 */

vector(30, n, binomial(3*n+1, n-1)) \\ Altug Alkan, Nov 04 2015

[binomial(3*n+1,n-1) for n in range(1,31)] # G. C. Greubel, Nov 09 2022

G.f.: (sqrt(4-27*x)*cos((2/3)*arcsin((3/2)*sqrt(3*x))) + sqrt(3*x)*sin((2/3)*arcsin((3/2)*sqrt(3*x))) - sqrt(4-27*x))/(3*sqrt(4-27*x)*x^2). - Emanuele Munarini, Oct 14 2014
From Peter Bala, Nov 04 2015: (Start)
With offset 0, the o.g.f. equals f(x)*g(x)^4, where f(x) is the o.g.f. for A005809 and g(x) is the o.g.f. for A001764.
More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(3*n + k,n). Cf. A045721 (k = 1), A025174 (k = 2), A004319 (k = 3), A013698 (k = 5), A165817 (k = -1), A117671 (k = -2). (End)
a(n) = [x^n] x/(1 - x)^(2*n+3). - Ilya Gutkovskiy, Oct 10 2017
From Karol A. Penson, Mar 02 2024: (Start)
G.f.: ((sqrt(3)*sqrt(x)*i + sqrt(4 - 27*x))*(4*sqrt(4 - 27*x) - 12*i*sqrt(3)*sqrt(x))^(2/3) + (-sqrt(3)*sqrt(x)*i + sqrt(4 - 27*x))*(4*sqrt(4 - 27*x) + 12*i*sqrt(3)*sqrt(x))^(2/3) - 8*sqrt(4 - 27*x))/(24*sqrt(4 - 27*x)*x^2), where i is the imaginary unit, i=sqrt(-1).
G.f.: hypergeometric3F2([5/3,2,7/3],[5/2,3],27*x/4).
G.f. = G satisfies the algebraic equation: 1 + (7*z-1)*G + (27*z-4)*z^2*G^2 + (27*z-4)*z^4*G^3 = 0. (End)
a(n) ~ 3^(3*n+3/2) / (2^(2*n+3) * sqrt(Pi*n)). - Amiram Eldar, Sep 05 2025

A270386 Expansion of (4/(3x/(1-x))) sin((1/3)arcsin(sqrt(27x/4/(1-x))))^2.

Original entry on oeis.org

1, 2, 9, 46, 256, 1510, 9283, 58848, 381963, 2525916, 16958498, 115288674, 792042589, 5490312864, 38352695246, 269719400974, 1908059370583, 13568804436340, 96942782340802, 695513575242284, 5008808999633736, 36195063931874308, 262372258663337954

Offset: 0

Vladimir Kruchinin, Mar 16 2016

nonn

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

Cf. A006013.

Table[Sum[(Binomial[n - 1, n - k]*((Binomial[3*k + 1, k + 1])/(2*k + 1))), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 16 2016, after Vladimir Kruchinin *)

a(n):=(sum(binomial(n-1,n-k)*((binomial(3*k+1,k+1))/(2*k+1)),k,0,n));

a(n) = sum(k=0, n, binomial(n-1,n-k)*(binomial(3*k+1,k+1)/(2*k+1))); \\ Michel Marcus, Mar 16 2016

a = lambda n: simplify(2*hypergeometric([5/3, 7/3, 1-n],[5/2, 3],-27/4)) if n>0 else 1
[a(n) for n in range(23)] # Peter Luschny, Mar 16 2016

a(n) = Sum_{k=0..n} binomial(n-1,n-k)*(binomial(3*k+1,k+1)/(2*k+1)).
G.f.: g(x/(1-x)) where g(x) is the g.f. of A006013.
a(n) ~ 31^(n + 1/2) / (sqrt(Pi) * n^(3/2) * 2^(2*n + 2)). - Vaclav Kotesovec, Mar 16 2016
a(n) = 2*hypergeometric([5/3, 7/3, 1-n], [5/2, 3], -27/4) for n>0. - Peter Luschny, Mar 16 2016
D-finite with recurrence: 2*(n+1)*(2*n+1)*a(n) +(-39*n^2+8*n+5)*a(n-1) +(66*n-37)*(n-2)*a(n-2) -31*(n-2)*(n-3)*a(n-3)=0. - R. J. Mathar, Jun 07 2016

A047000 Array T read by diagonals; T(h,k)=number of paths consisting of steps from (0,0) to (h,k) such that each step has length 1 directed up or right and no step touches the line y=x/2 except at the endpoints.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 2, 1, 1, 1, 4, 5, 2, 2, 1, 1, 5, 9, 7, 4, 3, 1, 1, 6, 14, 16, 7, 3, 4, 1, 1, 7, 20, 30, 23, 7, 7, 5, 1, 1, 8, 27, 50, 53, 30, 14, 12, 6, 1, 1, 9, 35, 77, 103, 83, 30, 12, 18, 7, 1, 1, 10, 44, 112, 180, 186, 113, 30, 30

Offset: 0

Clark Kimberling

Touches here includes the case where a step touches the line at a midpoint.

			Diagonals (starting on row #0): {1}; {1,1}; {1,1,1}; {1,2,2,1}; {1,3,2,1,1}; ...
T(2,3) = 5; the 5 allowed paths to (2,3) are UUURR, UURUR, UURRU, URUUR, and URURU.

The sequence T(2n, n)/2 for n=1, 2, 3, ... is A006013.

T(h,k)=if(h==0 || k==0,1,T(h-1,k)*(h-1!=2*k)+T(h,k-1)*(h!=2*k-2 && h!=2*k-1)) /* Inefficient. */

Definition corrected by Franklin T. Adams-Watters, Mar 10 2011

A092276 Triangle read by rows: T(n,k) is the number of noncrossing trees with root degree equal to k.

Original entry on oeis.org

1, 2, 1, 7, 4, 1, 30, 18, 6, 1, 143, 88, 33, 8, 1, 728, 455, 182, 52, 10, 1, 3876, 2448, 1020, 320, 75, 12, 1, 21318, 13566, 5814, 1938, 510, 102, 14, 1, 120175, 76912, 33649, 11704, 3325, 760, 133, 16, 1, 690690, 444015, 197340, 70840, 21252, 5313, 1078, 168, 18, 1

Offset: 1

Emeric Deutsch, Feb 24 2004

With offset 0, Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A006013. - Philippe Deléham, Jan 23 2010

			Triangle begins:
     1;
     2,    1;
     7,    4,    1;
    30,   18,    6,   1;
   143,   88,   33,   8,  1;
   728,  455,  182,  52, 10,  1;
  3876, 2448, 1020, 320, 75, 12, 1;
  ...
Top row of M^3 = (30, 18, 6, 1)
From _Peter Bala_, Nov 25 2024: (Start)
The transposed array as an infinite product of upper triangular arrays:
  /1 2 3 4 5 ... \/1            \/1              \       /1 2 7 30 143 ...\
  |  1 2 3 4 ... ||  1 2 3 4 ...||  1            |       |  1 4 18  88 ...|
  |    1 2 3 ... ||    1 2 3 ...||    1 2 3 4 ...| ... = |    1  6  33 ...|
  |      1 2 ... ||      1 2 ...||      1 2 3 ...|       |       1   8 ...|
  |        1 ... ||        1 ...||        1 2 ...|       |           1 ...|
  |          ... ||          ...||            ...|       |             ...|
Cf. A078812. (End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Peter Bala, Factorisations of some Riordan arrays as infinite products
Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
Paul Barry, The second production matrix of a Riordan array, arXiv:2011.13985 [math.CO], 2020.
P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.
M. Noy, Enumeration of noncrossing trees on a circle, Discrete Math., 180, 301-313, 1998.

Row sums give sequence A001764.

Columns 1..5 are A006013, A006629, A006630, A006631, A233657.

T := proc(n,k) if k=n then 1 else 2*k*binomial(3*n-k,n-k)/(3*n-k) fi end: seq(seq(T(n,k),k=1..n),n=1..11);

t[n_, n_] = 1; t[n_, k_] := 2*k*Binomial[3*n-k, n-k]/(3*n-k); Table[t[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 22 2012, after Maple *)

T(n, k) = 2*k*binomial(3*n-k, n-k)/(3*n-k); \\ Andrew Howroyd, Nov 06 2017

T(n, k) = 2*k*binomial(3n-k, n-k)/(3n-k).
G.f.: 1/(1-t*z*g^2), where g := 2*sin(arcsin(3*sqrt(3*z)/2)/3)/sqrt(3*z) is the g.f. of the sequence A001764.
T(n, k) = Sum_{j>=1} j*T(n-1, k-2+j). - Philippe Deléham, Sep 14 2005
With offset 0, T(n,k) = ((n+1)/(k+1))*binomial(3n-k+1, n-k). - Philippe Deléham, Jan 23 2010
From Gary W. Adamson, Jul 07 2011: (Start)
Let M = the production matrix
  2, 1;
  3, 2, 1;
  4, 3, 2, 1;
  5, 4, 3, 2, 1;
  ...
Top row of M^(n-1) generates n-th row terms of triangle A092276. Leftmost terms of each row = A006013 starting (1, 2, 7, 30, 143, ...). (End)
Working with an offset of 0, the inverse array is the Riordan array ((1 - x)^2, x*(1 - x)^2). - Peter Bala, Apr 30 2024

A036829 a(n) = Sum_{k=0..n-1} C(3k,k)C(3n-3k-2,n-k-1).

Original entry on oeis.org

0, 1, 7, 48, 327, 2221, 15060, 102012, 690519, 4671819, 31596447, 213633696, 1444131108, 9760401756, 65957919496, 445671648228, 3011064814455, 20341769686311, 137412453018933, 928188965638464, 6269358748632207, 42343731580741821

Offset: 0

N. J. A. Sloane

nonn

M. Petkovsek et al., A=B, Peters, 1996, p. 97.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A006256.
Cf. A001764, A006013, A007318, A036917, A062236.
Cf. A000302, A308523, A386367, A386368.

a036829 n = sum $ map
   (\k -> (a007318 (3*k) k) * (a007318 (3*n-3*k-2) (n-k-1))) [0..n-1]
-- Reinhard Zumkeller, May 24 2012

Table[Sum[Binomial[3k,k]Binomial[3n-3k-2,n-k-1],{k,0,n-1}],{n,0,30}] (* Harvey P. Dale, Jan 10 2012 *)

G.f.: (g-g^2)/(3*g-1)^2 where g*(1-g)^2 = x. - Mark van Hoeij, Nov 09 2011
Recurrence: 8*(n-1)*(2*n-1)*a(n) = 6*(36*n^2-81*n+49)*a(n-1) - 81*(3*n-5)*(3*n-4)*a(n-2). - Vaclav Kotesovec, Nov 19 2012
a(n) ~ 3^(3*n-1)/2^(2*n+1). - Vaclav Kotesovec, Dec 29 2012
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/3) * log( Sum_{k>=0} binomial(3*k,k)*x^k ). - Seiichi Manyama, Jul 19 2025
G.f.: (g-1)/(3-2*g)^2 where g=1+x*g^3. - Seiichi Manyama, Jul 26 2025

cp's OEIS Frontend

A117671 a(n) = binomial(3*n+1, n+1).

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A251573 E.g.f.: exp(3*x*G(x)^2) / G(x)^2 where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

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A118969 a(n) = 2*binomial(5*n+1,n)/(4*n+2).

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A130565 Member k=6 of a family of generalized Catalan numbers.

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A051255 Number of cyclically symmetric transpose complement plane partitions in a 2n X 2n X 2n box.

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A236194 a(n) = binomial(3n+1, n-1).

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A270386 Expansion of (4/(3*x/(1-x))) * sin((1/3)*arcsin(sqrt(27*x/4/(1-x))))^2.

A251573 E.g.f.: exp(3xG(x)^2) / G(x)^2 where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

A118969 a(n) = 2binomial(5n+1,n)/(4*n+2).

A270386 Expansion of (4/(3x/(1-x))) sin((1/3)arcsin(sqrt(27x/4/(1-x))))^2.

A036829 a(n) = Sum_{k=0..n-1} C(3k,k)C(3n-3k-2,n-k-1).