A000309 -id:A000309 - cp's OEIS frontend

A002005 Number of rooted planar cubic maps with 2n vertices.

1, 4, 32, 336, 4096, 54912, 786432, 11824384, 184549376, 2966845440, 48855252992, 820675092480, 14018773254144, 242919827374080, 4261707069259776, 75576645116559360, 1353050213048123392, 24428493151359467520, 444370175232646840320, 8138178004138611179520

Offset: 0

N. J. A. Sloane

nonn

Equivalently, number of rooted planar triangulations with 2n faces.

The September 2018 talk by Noam Zeilberger (see link to video) connects three topics (planar maps, Tamari lattices, lambda calculus) and eight sequences: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827. - N. J. A. Sloane, Sep 17 2018

R. C. Mullin, E. Nemeth and P. J. Schellenberg, The enumeration of almost cubic maps, pp. 281-295 in Proceedings of the Louisiana Conference on Combinatorics, Graph Theory and Computer Science. Vol. 1, edited R. C. Mullin et al., 1970.
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).

Gheorghe Coserea, Table of n, a(n) for n = 0..1000
Valentin Bonzom, Guillaume Chapuy, Maciej Dolega, Enumeration of non-oriented maps via integrability, Alg. Combin. 5 (6) (2022) p 1363-1390, A.3
Mireille Bousquet-Mélou, Counting planar maps, coloured or uncoloured, 23rd British Combinatorial Conference, Jul 2011, Exeter, United Kingdom. 392, pp.1-50, 2011, London Math. Soc. Lecture Note Ser., hal-00653963. See p.13.
Evgeniy Krasko and Alexander Omelchenko, Enumeration of r-regular maps on the torus. Part I: Rooted maps on the torus, the projective plane and the Klein bottle. Sensed maps on the torus, Discrete Mathematics (2019) Vol. 342, Issue 2, 584-599. Also arXiv:1709.03225 [math.CO].
Maxim Krikun, Explicit enumeration of triangulations with multiple boundaries, arXiv:0706.0681 [math.CO], 2007. [Comment from _Gheorghe Coserea_, Dec 26 2015: the formula in the paper for almost trivalent maps is 2 * 4^(k-1) * (3k)!!/ ((k+1)!*(k+2)!!); however, the exponent of 4 should be k not (k-1) i.e. 2 * 4^k * (3k)!! / ((k+1)!*(k+2)!!)]
Noam Zeilberger, A theory of linear typings as flows on 3-valent graphs, arXiv:1804.10540 [cs.LO], 2018.
Noam Zeilberger, A Sequent Calculus for a Semi-Associative Law, arXiv preprint 1803.10030 [math.LO], March 2018 (A revised version of a 2017 conference paper).
Noam Zeilberger, A proof-theoretic analysis of the rotation lattice of binary trees, Part 1 (video), Part 2, Rutgers Experimental Math Seminar, Sep 13 2018.
Jian Zhou, Fat and Thin Emergent Geometries of Hermitian One-Matrix Models, arXiv:1810.03883 [math-ph], 2018.

Sequences mentioned in the Noam Zeilberger 2018 video: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827.
Column k=0 of A266240.
Cf. A000217, A358367.

seq(2*8^n*binomial(n*3/2, n)/((n + 2)*(n + 1)), n = 0..19); # Peter Luschny, Nov 14 2022

Table[2^(2 n + 1) (3 n)!!/((n + 2)! n!!), {n, 0, 20}] (* Vincenzo Librandi, Dec 28 2015 *)
CoefficientList[Series[(-1 + 96 z + Hypergeometric2F1[-2/3,-1/3,1/2,432z^2]- 96 z Hypergeometric2F1[-1/6,1/6,3/2,432z^2])/(192 z^2), {z, 0, 10}], z] (* Benedict W. J. Irwin, Aug 07 2016 *)

factorial2(n) = my(x = (2^(n\2)*(n\2)!)); if (n%2, n!/x, x);
a(n) = 2^(2*n+1)*factorial2(3*n)/((n+2)!*factorial2(n));
vector(20, i, a(i-1))
\\ test: y = Ser(vector(201, n, a(n-1))); x*(1-432*x^2)*y' == 64*x^2*y^2 + (288*x^2 - 64*x - 1)*y + 72*x + 1
\\ Gheorghe Coserea, Jun 13 2017

a(n) = 2^(2*n+1)*(3*n)!!/((n+2)!*n!!). - Sean A. Irvine, May 19 2013
a(n) ~ sqrt(6/Pi) * n^(-5/2) * (12*sqrt(3))^n. - Gheorghe Coserea, Feb 25 2016
G.f.: (96*x - 1 + 2F1(-2/3, -1/3; 1/2; 432*x^2) - 96*x*2F1(-1/6, 1/6; 3/2; 432*x^2))/(192*x^2). - Benedict W. J. Irwin, Aug 07 2016
From Gheorghe Coserea, Jun 13 2017: (Start)
G.f. y(x) satisfies:
x*(1-432*x^2)*deriv(y,x) = 64*x^2*y^2 + (288*x^2 - 64*x - 1)*y + 72*x + 1.
0 = 64*x^3*y^3 + x*(1-96*x)*y^2 + (30*x-1)*y - 27*x + 1.
(End).
D-finite with recurrence (n+2)*(n+1)*a(n) -48*(3*n-2)*(3*n-4)*a(n-2)=0. - R. J. Mathar, Feb 08 2021
From Karol A. Penson and Katarzyna Gorska (katarzyna.gorska@ifj.edu.pl), Nov 02 2022: (Start)
a(n) = Integral_{x=0..12*sqrt(3)} x^n*W(x), where
W(x) = (T1(x) + T2(x)) / T3(x), and
T1(x) = -x^(2/3) * (108 + sqrt(3) * sqrt(432 - x^2));
T2(x) = 3^(1/6)*(36+sqrt(3)*sqrt(432-x^2))^(2/3) * (-432+x^2+36*sqrt(3)* sqrt(432-x^2)) / sqrt(432-x^2);
T3(x) = (128*3^(5/6)*Pi*x^(1/3)*(36+sqrt(3)*sqrt(432-x^2))^(1/3)).
This integral representation is unique as W(x) is the solution of the Hausdorff power moment problem. Using only the definition of a(n), W(x) can be proven to be positive. W(x) is singular at x = 0, with the singularity x^(-1/3), and for x > 0 is monotonically decreasing to zero at x = 12*sqrt(3). (End)
a(n) = 2^(3*n + 1)*binomial(n*3/2, n)/((n + 1)*(n + 2)) = A358367(n) / A000217(n + 1). - Peter Luschny, Nov 14 2022

More terms from Sean A. Irvine, May 19 2013

A006335 a(n) = 4^n(3n)!/((n+1)!(2n+1)!).

Original entry on oeis.org

1, 2, 16, 192, 2816, 46592, 835584, 15876096, 315031552, 6466437120, 136383037440, 2941129850880, 64614360416256, 1442028424527872, 32619677465182208, 746569714888605696, 17262927525017812992, 402801642250415636480, 9474719710174783733760, 224477974671833337692160

Offset: 0

N. J. A. Sloane

Number of planar lattice walks of length 3n starting and ending at (0,0), remaining in the first quadrant and using only NE,W,S steps.
Equals row sums of triangle A140136. - Michel Marcus, Nov 16 2014
Number of linear extensions of the poset V x [n], where V is the 3-element poset with one least element and two incomparable elements: see Kreweras and Niederhausen (1981) and Hopkins and Rubey (2020) references. - Noam Zeilberger, May 28 2020

			G.f. = 1 + 2*x + 16*x^2 + 192*x^3 + 2816*x^4+ 46592*x^5 + 835584*x^6 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..700
Andrei Asinowski, Cyril Banderier, and Sarah J. Selkirk, From Kreweras to Gessel: A walk through patterns in the quarter plane, Séminaire Lotharingien de Combinatoire, Proc. 35th Conf. Formal Power Series and Alg. Comb. (Davis, 2023) Vol. 89B, Art. #30.
Olivier Bernardi, Bijective counting of Kreweras walks and loopless triangulations, Journal of Combinatorial Theory, Series A 114:5 (2007), 931-956.
M. Bousquet-Mélou, Walks in the quarter plane: Kreweras' algebraic model, arXiv:math/0401067 [math.CO], 2004-2006.
M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008.
Sam Hopkins and Martin Rubey, Promotion of Kreweras words, arXiv:2005.14031 [math.CO], 2020.
G. Kreweras, Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, 6 (1965), circa p. 82.
G. Kreweras and H. Niederhausen, Solution of an enumerative problem connected with lattice paths, European J. Combin., 2 (1981), 55-60.

Equals 2^(n-1) * A000309(n-1) for n>1.
Cf. A098272. First row of array A098273.
Column of A176129, A214631, A214722, A340591.

[4^n*Factorial(3*n)/(Factorial(n+1)*Factorial(2*n+1)) : n in [0..20]]; // Wesley Ivan Hurt, Nov 16 2014

A006335:=n->4^n*(3*n)!/((n+1)!*(2*n+1)!): seq(A006335(n), n=0..20); # Wesley Ivan Hurt, Nov 16 2014

aux[i_Integer, j_Integer, n_Integer] := Which[Min[i, j, n] < 0 || Max[i, j] > n, 0, n == 0, KroneckerDelta[i, j, n], True, aux[i, j, n] = aux[-1 + i, j, -1 + n] + aux[i, -1 + j, -1 + n] + aux[1 + i, 1 + j, -1 + n]]; Table[aux[0, 0, 3 n], {n, 0, 25}] (* Manuel Kauers, Nov 18 2008 *)
Table[(4^n (3 n)! / ((n + 1)! (2 n + 1)!)), {n, 0, 200}] (* Vincenzo Librandi, Nov 17 2014 *)

{a(n) = if( n<0, 0, 4^n * (3*n)! / ((n+1)! * (2*n+1)!))}; /* Michael Somos, Jan 23 2003 */

def a(n):
    return (4**n * binomial(3 * n, 2 * n)) // ((n + 1) * (2 * n + 1))
# F. Chapoton, Jun 01 2020

G.f.: (1/(12*x)) * (hypergeom([ -2/3, -1/3],[1/2],27*x)-1). - Mark van Hoeij, Nov 02 2009
a(n+1) = 6*(3*n+2)*(3*n+1)*a(n)/((2+n)*(2*n+3)). - Robert Israel, Nov 17 2014
a(n) ~ 3^(3*n + 1/2) / (4*sqrt(Pi)*n^(5/2)). - Vaclav Kotesovec, Mar 26 2016
E.g.f.: 2F2(1/3,2/3; 3/2,2; 27*x). - Ilya Gutkovskiy, Jan 25 2017

Edited by N. J. A. Sloane, Dec 20 2008 at the suggestion of R. J. Mathar

A153231 a(n) = 2^n * binomial(3n,n)/(2n+1).

Original entry on oeis.org

1, 2, 12, 96, 880, 8736, 91392, 992256, 11075328, 126297600, 1465052160, 17233182720, 205074874368, 2464404045824, 29864206663680, 364535993597952, 4477993284993024, 55316387638149120, 686720560048373760, 8563155161736806400, 107206525476085432320

Offset: 0

Yidong Sun (sydmath(AT)yahoo.com.cn), Dec 21 2008

a(n) is also the number of rooted generalized noncrossing trees on n+1 vertices.
The series reversion of y = x +2*x^3 is x = y -2*y^3 +12*y^5 -96*y^7 +880*y^9 -8736*y^11 +... - R. J. Mathar, Sep 29 2012
Lattice paths in the 1st quadrant from (0,0) to (3n,0) using steps D(1,-1) and two types of U(1,2). - David Scambler, Jun 22 2013
From Torsten Muetze, May 08 2024: (Start)
a(n) also counts ternary trees with n nodes that are colored red or blue.
a(n) also counts triangulations of a convex (2n+2)-gon whose points are colored red and blue alternatingly, and that do not have monochromatic triangles (i.e., every triangle has at least one red point and at least one blue point). (End)

Bruce E. Sagan, Proper partitions of a polygon and k-Catalan numbers, Ars Combinatoria, 88 (2008), 109-124.

Michael De Vlieger, Table of n, a(n) for n = 0..889
CombOS - Combinatorial Object Server, Generate k-ary trees and dissections
Hsien-Kuei Hwang, Mihyun Kang, and Guan-Huei Duh, Asymptotic Expansions for Sub-Critical Lagrangean Forms, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.
Bruce E. Sagan, Proper partitions of a polygon and k-Catalan numbers, arXiv:math/0407280 [math.CO], 2004.
Anssi Yli-Jyrä and Carlos Gómez-Rodríguez, Generic Axiomatization of Families of Noncrossing Graphs in Dependency Parsing, arXiv:1706.03357 [cs.CL], 2017.

Cf. A000309, A001764.

Cf. A369510 (colorful triangulations with an odd number of points).

[2^n*Binomial(3*n,n)/(2*n+1): n in [0..30]]; // Vincenzo Librandi, Sep 08 2015

Table[2^n Binomial[3n, n]/(2n+1), {n, 0, 25}] (* Vincenzo Librandi, Sep 08 2015 *)

a(n) = 2^n*binomial(3*n,n)/(2*n+1); \\ Altug Alkan, Sep 24 2018

[2^n*binomial(3*n,n)/(2*n+1) for n in range(31)] # G. C. Greubel, Mar 08 2023

a(n) = 2^n*A001764(n). - R. J. Mathar, Oct 06 2012
D-finite with recurrence n*(2*n+1)*a(n) -3*(3*n-1)*(3*n-2)*a(n-1) = 0. - R. J. Mathar, Nov 16 2012
a(n) = (n+1)*A000309(n). - Johannes W. Meijer, Aug 22 2013
G.f.: sqrt(2)/sqrt(3*x)*sin(1/3*asin(sqrt(27*x/2))). - Vladimir Kruchinin, Sep 08 2015
E.g.f.: Hypergeometric2F2(1/3,2/3; 1,3/2; 27*x/2). - Ilya Gutkovskiy, Nov 23 2017

More terms from N. J. A. Sloane, Dec 21 2008

A000264 Number of 3-edge-connected rooted cubic maps with 2n nodes and a distinguished Hamiltonian cycle.

Original entry on oeis.org

1, 1, 3, 14, 80, 518, 3647, 27274, 213480, 1731652, 14455408, 123552488, 1077096124, 9548805240, 85884971043, 782242251522, 7203683481720, 66989439309452, 628399635777936, 5940930064989720, 56562734108608536

Offset: 1

N. J. A. Sloane

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).

Vincenzo Librandi, Table of n, a(n) for n = 1..200
L. B. Richmond, On Hamiltonian polygons, J. Combinatorial Theory Ser. B 21 (1976), no. 1, 81--87. MR0432491 (55 #5479) [See v_n].
W. T. Tutte, A census of Hamiltonian polygons, Canad. J. Math., 14 (1962), 402-417.

Cf. A000309, A000356, A004304.

max = 21; b[n_] := (2n)!*(2n + 2)!/(2*n!*(n + 1)!^2*(n + 2)!); b[0] = 0; bf[x_] := Sum[b[n]*x^n, {n, 0, max}]; Clear[a]; a[0] = 0; a[1] = a[2] = 1; af[x_] := Sum[a[n]*x^n, {n, 0, max}]; se = Series[bf[x] - af[x*(1 + 2*bf[x])^2], {x, 0, max}] // Normal; Table[a[n], {n, 1, max}] /. SolveAlways[se == 0, x] // First (* Jean-François Alcover, Jan 31 2013, after Sean A. Irvine *)

Let b(n)=(2n)!*(2n+2)!/(2*n!*(n+1)!^2*(n+2)!). Let B(x) be the generating function producing b(n), and A(x) be the generating function producing a(n). Then these sequences satisfy the functional equation B(x)=A(x(1+2*B(x))^2). - Sean A. Irvine, Apr 05 2010

Better definition from Michael Albert, Oct 24 2008

More terms from Sean A. Irvine, Apr 05 2010

A218473 Number of 3n-length 3-ary words, either empty or beginning with the first letter of the alphabet, that can be built by repeatedly inserting triples of identical letters into the initially empty word.

Original entry on oeis.org

1, 1, 7, 61, 591, 6101, 65719, 729933, 8297247, 96044101, 1128138567, 13411861629, 161066465583, 1950996039669, 23808159962839, 292413627476141, 3611870017079871, 44838216520062117, 559127724970143079, 7000374603097246173, 87964883375131331151

Offset: 0

Alois P. Heinz, Oct 29 2012

Alois P. Heinz, Table of n, a(n) for n = 0..300

Column k=3 of A213027. Cf. A000139, A000309, A001764.

a:= n-> `if`(n=0, 1, add(binomial(3*n, j)*(n-j)*2^j, j=0..n-1)/n):
seq(a(n), n=0..20);
# second Maple program
a:= proc(n) a(n):= `if`(n<3, [1, 1, 7][n+1], (-81*(3*n-1)*(3*n-5)*a(n-2)
       +(81*n^2-81*n+15)*a(n-1))/ ((2*n-1)*n))
    end:
seq(a(n), n=0..20);

Flatten[{1,Table[1/n*Sum[Binomial[3*n,j]*(n-j)*2^j,{j,0,n-1}],{n,1,20}]}] (* Vaclav Kotesovec, May 22 2013 *)
Flatten[{1,Table[FullSimplify[SeriesCoefficient[(1/(81*x-3)+2/((3-81*x)*(1-27*x-3*Sqrt[3*x*(27*x-2)])^(2/3))),{x,0,n}]],{n,1,10}]}] (* Vaclav Kotesovec, Jul 06 2013 *)

a(n) = (1/n) * Sum_{j=0..n-1} binomial(3*n,j)*(n-j)*2^j for n>0, a(0) = 1.
a(n) ~ 3^(3*n-3/2)/(sqrt(Pi)*2^(n-1)*n^(3/2)). - Vaclav Kotesovec, May 22 2013
G.f. (for n>0): (1/(81*x-3)+2/((3-81*x)*(1-27*x-3*sqrt(3*x*(27*x-2)))^(2/3))). - Vaclav Kotesovec, Jul 06 2013
From Peter Bala, Feb 06 2022: (Start)
The o.g.f. A(x) satisfies the algebraic equation 8*x - 36*x*A(x) + (54*x - 1)*A(x)^2 + (-27*x + 1)*A(x)^3 = 0.
A(x) = (6 - 4*T(2*x))/(2*T(2*x)^2 - 9*T(2*x) + 9), where T(x) = 1 + x*T(x)^3 is the o.g.f. of A001764.
A(x) = 1 + 2*x*B'(2*x)/B(2*x), where B(x) = 2 + x + 2*x^2 + 6*x^3 + 22*x^4 + 91*x^5 + ... is the o.g.f. of A000139.
exp(Sum_{n >= 1} a(n)*x*n/n) = 1 + x + 4*x^2 + 24*x^3 + 176*x^4 + 1456*x^5 + ... is the o.g.f. of A000309, a power series with integral coefficients. It follows that the Gauss congruences a(n*p^k) == a(n*p*(k-1)) (mod p^k) hold for all prime p and positive integers n and k. (End)

A358091 Triangle read by rows. Coefficients of the polynomials P(n, x) = 2^(n-2)(3n-1)* hypergeometric([-3*n, 1 - n, -n + 4/3], [-n, -n + 1/3], x). T(n, k) = [x^k] P(n, x).

Original entry on oeis.org

1, 5, -6, 16, -60, 48, 44, -288, 660, -440, 112, -1056, 4032, -7280, 4368, 272, -3360, 17952, -52224, 81600, -45696, 640, -9792, 67200, -267520, 656640, -930240, 496128, 1472, -26880, 225216, -1133440, 3740352, -8160768, 10767680, -5537664

Offset: 1

Peter Luschny, Oct 28 2022

			[1]    1;
[2]    5,     -6;
[3]   16,    -60,     48;
[4]   44,   -288,    660,     -440;
[5]  112,  -1056,   4032,    -7280,    4368;
[6]  272,  -3360,  17952,   -52224,   81600,   -45696;
[7]  640,  -9792,  67200,  -267520,  656640,  -930240,   496128;
[8] 1472, -26880, 225216, -1133440, 3740352, -8160768, 10767680, -5537664;

Cf. A062236, A000309.

def P(n):
    h = 2^(n-2)*(3*n-1)*hypergeometric([-3*n, 1 - n, -n + 4/3], [-n, -n + 1/3], x)
    return h.series(x, n+1).polynomial(SR)
for n in range(1, 9): print(P(n).list())
# To evaluate the polynomials use:
def p(n, t): return Integer(P(n)(x=t).n())
# For example the next statements yield A062236 and A000309.
print([p(n, -1/2) for n in range(1, 21)])
print([(-1)^n*p(n + 1, 1) for n in range(0, 22)])

P(n, -1/2) = A062236(n).

(-1)^n*P(n + 1, 1) = A000309(n).

A336110 Irregular triangle of Catalan-based numbers, read by rows.

Original entry on oeis.org

1, 1, 2, -2, 5, -14, 5, 14, -74, 74, -14, 42, -352, 668, -352, 42, 132, -1588, 4808, -4808, 1588, -132, 429, -6946, 30371, -48540, 30371, -6946, 429, 1430, -29786, 176270, -407810, 407810, -176270, 29786, -1430

Offset: 1

Sergii Voloshyn, Jul 08 2020

Calculation of the sum over the partitions of r of products of dimensions of two different representations of a symmetric group S_r gives
Sum_{L |- S_r} f(L)*f(l+q^N) = (r+q^N)! * G[N+1] * G[q+1]/(G[N+q+1]) * B_r(1!c_1, ..., r!c_r) where f(L) is the dimension of the symmetric group S_r, G[x] is Barnes function, and B_r() is the complete exponential Bell polynomial.
In the limit N -> infinity the coefficients [are?]
  c_1 = 1/(1+x), c_i = 1/(i*N^(2*(i-1)))*P(i-1), for i >= 2.
Coefficient of x^n in the numerator of P(i) is T(s, i).
This triangle of coefficients was discovered by Borisenko et al. In mathematical physics these coefficients appear as an important ingredient of series that define the free energy of the SU(N) standard lattice model in the large N limit.
They are easily obtained from the g.f.
Some special cases are given by A000108 (first column of the triangle), A138156 (second column of the triangle).
The sum of the numbers in row 2*k+1 is (-1)^k * A000260(k) * 2^(2*k).

			     1;
     1;
     2,     -2;
     5,    -14,      5;
    14,    -74,     74,     -14;
    42,   -352,    668,    -352,     42;
   132,  -1588,   4808,   -4808,   1588,    -132;
   429,  -6946,  30371,  -48540,  30371,   -6946,   429;
  1430, -29786, 176270, -407810, 407810, -176270, 29786, -1430;
  ...

Sergii Voloshyn, Table of n, a(n) for n = 1..120
O. Borisenko, V. Chelnokov, and Sergii Voloshyn, The large N limit of SU(N) integrals in lattice, arXiv:2008.00773 [hep-lat], 2020. See formula (48).
F. Green and S. Samuel, Calculating the large-N phase transition in gauge and matrix models, Nuclear Physics B 194, Issue 1, 11 January 1982, Pages 107-156, See Appendix A.

Cf. A000108, A000309, A138156, A000260.

T[L_, n_] := CatalanNumber[L] Sum[u[L, a]/u[0, a] M[n - 1 - 2*a, L], {a, 0, (n - 1)/2}]-4^L Sum[v[L, a]/v[0, a] M[n - 2 - 2*a, L], {a,0,(n-2)/2}];
M[n_, l_] := Sum[k!/n! BellY[n,k,Table[(-1)^(j-1) j!Binomial[3l+j,j], {j,n}]], {k, 0, n}];
u[k_, n_] := Product[Binomial[2 k + 2 l + 1, 3], {l, 1, n}];
v[k_, n_] := Product[ Binomial[2 k + 2 l + 2, 3], {l, 1, n}];
(* alternate program using coefficients in numerator *)
P[s_] := x CatalanNumber[s] HypergeometricPFQ[{s + 1/2, s + 1, s + 3/2}, {1/2, 3/2}, x^2] - x^2*4^s HypergeometricPFQ[{s + 1, s + 3/2, s + 2}, {3/2, 2}, x^2];
Table[CoefficientList[P[s] // Together // Numerator, x] // Rest, {s, 0, 10}] // Flatten (* amended by Jean-François Alcover, Sep 25 2020 *)
(* another program using coefficients in numerator *)
Needs["Combinatorica`"];
OA[p_,x_]:= (2^p(-(1/(x+1)))^(2p+1))/((2p+1)!(p+1)!) Sum[(-x/(1+x))^(p-r+1)Product[Pochhammer[1+Plus@@Table[3*k[[i]]-1,{i,1,j-1}],3*k[[j]]],{j,1,r}],{r, 1, p},{k,Compositions[p-r,r]+1}]; Table[CoefficientList[OA[s, x] // Together // Numerator, x] //
   Rest, {s, 0, 10}] // Flatten (* Sergii Voloshyn, Sep 03 2021 *)

Coefficients of x^n in the numerator of P(s) = (x * C[s]* 3F2[ s+ 1/2, s+1, s+3/2; 1/2,3/2; x^2] - x^2 * 4^s * 3F2[ s+1,s+3/2, s+2; 3/2, 2; x^2]), where C(s) are Catalan numbers.
or in a more explicit way (only for k >= 1)
T(s, n) = C(s) * U(s, n) - 4^s * V(s, n), where
U(s, n) = Sum_{a=0..(n-1)/2} (u(s, a)/u(0,a)) * M(s, n-1- 2a),
V(s, n) = Sum_{a=0..(n-2)/2} (v(s, a)/v(0,a)) * M(s, n-2- 2a), and
u(s, n) = Product_{L=1..n} binomial(2s+2L+1, 3),
v(s, n) = Product_{L=1..n} binomial(2s+2L+2, 3), and
M(m, n) = Sum_{L=1..n} L!/n! B_{n,l} ( x_1, ..., x_{n-L+1}), and
x_i = (-1)^{1+i} * (3s+i)_i = (-1)^{1+i} * i! * binomial(3s + i, i), where
B_{n,l} (x_1, ..., x_{n-L+1}) is the n-th partial or incomplete exponential Bell polynomial with monomials sorted into graded lexicographic order.
Sum of numbers in the particular row:
Sum_{n=1..2*k+1} T(2*k+1, n) = 2*(4*k+1)!/((k+1)!*(3*k+2)!) *2^(2*k) (odd s);
Sum_{n=1..2*k} T(2*k, n) = 0 (even s).
From Sergii Voloshyn, Oct 22 2020: (Start)
Formulae for particular columns:
T(s, 1) = C(s);
T(s, 2) = C(s)*(3*s+1) - 4^s;
T(s, 3) = C(s)*(binomial(2*s+3,3) + (3*s+1)^2 - binomial(3*s +2,2)) - 4^s*(3*s+1);
T(s, 4) = C(s)*((2*s+1)binomial(2*s+3,3) +(3*s+1)^3 - 2(3*s+1)* binomial(3*s+2,2)+ binomial(3*s+3, 3)) - 4^s*(binomial(3*s+4, 3)/4 + (3*s+1)^2 - binomial(3*s+2,2));
...
T(s, s) = (-1)^(s+1)*C(s). (End)
From Sergii Voloshyn, Mar 17 2021: (Start)
Recursion ( P[0] = x/(1+x) ):
x*(d^3/dx^3)*P[s] = (1/4)*(2*k+2)*(2*k+3)*(2*k+4)*P[s+1]
for P[s_] := x CatalanNumber[s] HypergeometricPFQ[{s + 1/2, s + 1, s + 3/2}, {1/2, 3/2}, x^2] - x^2*4^s HypergeometricPFQ[{s + 1, s + 3/2, s + 2}, {3/2, 2}, x^2].
(End)
Recursion for array ( T(1,1) = 1 ): T(k, p) = (1/(k*(k+1)*(2k+1)))*[p*(p+1)*(p+2)*T(k-1,p+2) - 3*p*(p+1)*(3*k-p-1)*T(k-1,p+1) + 3*p*(3*k-p)*(3*k-p-1)*T(k-1,p) - (3*k-p+1)*(3*k-p)*(3*k-p-1)*T(k-1,p-1)]; p =[1,...,k]. - Sergii Voloshyn, Apr 14 2021
From Sergii Voloshyn, Apr 17 2021: (Start)
G.f.: Sum_{m>=1} x^m Om(k, m) = (1/(1+x)^(3*k+1))*Sum_{n=1..k} x^k * T(k,n);
Om(k, m) = (12^k/((1+k)!*(2k+1)!)) * Product_{L=1..k} binomial(m+2L, 3).
(End)
From Sergii Voloshyn, Apr 25 2021: (Start)
Differential equation for P[k_]:
x*(x^2-1)*(d^3/dx^3)P[s] + (2*k+2)*3*x^2*(d^2/dx^2)P[s] +(2*k+2)*(2*k+1)*3*x (d/dx)P[s] + (2*k+2)(2*k+1)*2*k*P[s] = 0.
Discrete set equation for T(k,n) (n=-1..k-2) at fixed k:
(k-n-1)*(k-n)*(k-n+1)*T(k,n) - (k-n-1)*(k-n)*(8*k+n+5)*T(k,n+1) - (n+1)*(n+2)*(9-n+3)*T(k,n+2) + (n+1)*(n+2)*(n+3)*T(k,n+3) = 0
and
Sum_{m=1..k} (k*(5+7*k) + 12*n*(n-1-k))*T(k,n) = 0. (End)
P[k_]:= (2^k)/((2*k+1)!*(k+1)!)*(-1/(1+x))^{2k+1}[Sum_{r(1)+...+ r(L)=k} (-x/(1+x))^{k-L+1}* 1^{(3*r(1))}*(1+3*r(1)-1)^{(3*r(2))}*... *(1+Sum_{i=1..L-1} 3 r(i) -L+1)^{(3*r(L))}] where Sum_ r_i = k runs over all integer compositions of k, L is a number of parts of this composition and 1^{(3 r(1))} is a rising factorial. - Sergii Voloshyn, Sep 03 2021
Sum_{k} abs(T(n,k)) = A000309(n-1). - Sergii Voloshyn, Nov 20 2024

cp's OEIS Frontend

A002005 Number of rooted planar cubic maps with 2n vertices.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A006335 a(n) = 4^n*(3*n)!/((n+1)!*(2*n+1)!).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A153231 a(n) = 2^n * binomial(3n,n)/(2n+1).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

Extensions

A000264 Number of 3-edge-connected rooted cubic maps with 2n nodes and a distinguished Hamiltonian cycle.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A218473 Number of 3n-length 3-ary words, either empty or beginning with the first letter of the alphabet, that can be built by repeatedly inserting triples of identical letters into the initially empty word.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A358091 Triangle read by rows. Coefficients of the polynomials P(n, x) = 2^(n-2)*(3*n-1)* hypergeometric([-3*n, 1 - n, -n + 4/3], [-n, -n + 1/3], x). T(n, k) = [x^k] P(n, x).

Views

Author

Keywords

Examples

Crossrefs

Programs

SageMath

Formula

A336110 Irregular triangle of Catalan-based numbers, read by rows.

Views

Author

A006335 a(n) = 4^n(3n)!/((n+1)!(2n+1)!).

A358091 Triangle read by rows. Coefficients of the polynomials P(n, x) = 2^(n-2)(3n-1)* hypergeometric([-3*n, 1 - n, -n + 4/3], [-n, -n + 1/3], x). T(n, k) = [x^k] P(n, x).