This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A033282 #254 Jul 23 2025 14:36:25
%S A033282 1,1,2,1,5,5,1,9,21,14,1,14,56,84,42,1,20,120,300,330,132,1,27,225,
%T A033282 825,1485,1287,429,1,35,385,1925,5005,7007,5005,1430,1,44,616,4004,
%U A033282 14014,28028,32032,19448,4862,1,54,936,7644,34398,91728,148512,143208,75582,16796
%N A033282 Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.
%C A033282 T(n+3, k) is also the number of compatible k-sets of cluster variables in Fomin and Zelevinsky's cluster algebra of finite type A_n. Take a row of this triangle regarded as a polynomial in x and rewrite as a polynomial in y := x+1. The coefficients of the polynomial in y give a row of the triangle of Narayana numbers A001263. For example, x^2 + 5*x + 5 = y^2 + 3*y + 1. - _Paul Boddington_, Mar 07 2003
%C A033282 Number of standard Young tableaux of shape (k+1,k+1,1^(n-k-3)), where 1^(n-k-3) denotes a sequence of n-k-3 1's (see the Stanley reference).
%C A033282 Number of k-dimensional 'faces' of the n-dimensional associahedron (see Simion, p. 168). - _Mitch Harris_, Jan 16 2007
%C A033282 Mirror image of triangle A126216. - _Philippe Deléham_, Oct 19 2007
%C A033282 For relation to Lagrange inversion or series reversion and the geometry of associahedra or Stasheff polytopes (and other combinatorial objects) see A133437. - _Tom Copeland_, Sep 29 2008
%C A033282 Row generating polynomials 1/(n+1)*Jacobi_P(n,1,1,2*x+1). Row n of this triangle is the f-vector of the simplicial complex dual to an associahedron of type A_n [Fomin & Reading, p. 60]. See A001263 for the corresponding array of h-vectors for associahedra of type A_n. See A063007 and A080721 for the f-vectors for associahedra of type B and type D respectively. - _Peter Bala_, Oct 28 2008
%C A033282 f-vectors of secondary polytopes for Grobner bases for optimization and integer programming (see De Loera et al. and Thomas). - _Tom Copeland_, Oct 11 2011
%C A033282 From Devadoss and O'Rourke's book: The Fulton-MacPherson compactification of the configuration space of n free particles on a line segment with a fixed particle at each end is the n-Dim Stasheff associahedron whose refined f-vector is given in A133437 which reduces to A033282. - _Tom Copeland_, Nov 29 2011
%C A033282 Diagonals of A132081 are rows of A033282. - _Tom Copeland_, May 08 2012
%C A033282 The general results on the convolution of the refined partition polynomials of A133437, with u_1 = 1 and u_n = -t otherwise, can be applied here to obtain results of convolutions of these polynomials. - _Tom Copeland_, Sep 20 2016
%C A033282 The signed triangle t(n, k) =(-1)^k* T(n+2, k-1), n >= 1, k = 1..n, seems to be obtainable from the partition array A111785 (in Abramowitz-Stegun order) by adding the entries corresponding to the partitions of n with the number of parts k. E.g., triangle t, row n=4: -1, (6+3) = 9, -21, 14. - _Wolfdieter Lang_, Mar 17 2017
%C A033282 The preceding conjecture by Lang is true. It is implicit in Copeland's 2011 comments in A086810 on the relations among a gf and its compositional inverse for that entry and inversion through A133437 (a differently normalized version of A111785), whose integer partitions are the same as those for A134685. (An inversion pair in Copeland's 2008 formulas below can also be used to prove the conjecture.) In addition, it follows from the relation between the inversion formula of A111785/A133437 and the enumeration of distinct faces of associahedra. See the MathOverflow link concernimg Loday and the Aguiar and Ardila reference in A133437 for proofs of the relations between the partition polynomials for inversion and enumeration of the distinct faces of the A_n associahedra, or Stasheff polytopes. - _Tom Copeland_, Dec 21 2017
%C A033282 The rows seem to give (up to sign) the coefficients in the expansion of the integer-valued polynomial (x+1)*(x+2)^2*(x+3)^2*...*(x+n)^2*(x+n+1)/(n!*(n+1)!) in the basis made of the binomial(x+i,i). - _F. Chapoton_, Oct 07 2022
%C A033282 Chapoton's observation above is correct: the precise expansion is (x+1)*(x+2)^2*(x+3)^2*...*(x+n)^2*(x+n+1)/ (n!*(n+1)!) = Sum_{k = 0..n-1} (-1)^k*T(n+2,n-k-1)*binomial(x+2*n-k,2*n-k), as can be verified using the WZ algorithm. For example, n = 4 gives (x+1)*(x+2)^2*(x+3)^2*(x+4)^2*(x+5)/(4!*5!) = 14*binomial(x+8,8) - 21*binomial(x+7,7) + 9*binomial(x+6,6) - binomial(x+5,5). - _Peter Bala_, Jun 24 2023
%D A033282 S. Devadoss and J. O'Rourke, Discrete and Computational Geometry, Princeton Univ. Press, 2011 (See p. 241.)
%D A033282 Ronald L. Graham, Donald E. Knuth, Oren Patashnik, Concrete Mathematics, 2nd ed., Addison-Wesley, 1994. Exercise 7.50, pages 379, 573.
%D A033282 T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Section 5.8.
%H A033282 Vincenzo Librandi, <a href="/A033282/b033282.txt">Table of n, a(n) for n = 3..2000</a>
%H A033282 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Barry/barry91.html">On Integer-Sequence-Based Constructions of Generalized Pascal Triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
%H A033282 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Barry3/barry93.html">Continued fractions and transformations of integer sequences</a>, JIS 12 (2009) 09.7.6.
%H A033282 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Barry3/barry252.html">On the Inverses of a Family of Pascal-Like Matrices Defined by Riordan Arrays</a>, Journal of Integer Sequences, 16 (2013), #13.5.6.
%H A033282 Paul Barry, <a href="https://arxiv.org/abs/1803.06408">Three Études on a sequence transformation pipeline</a>, arXiv:1803.06408 [math.CO], 2018.
%H A033282 Paul Barry, <a href="https://arxiv.org/abs/1805.02274">On the f-Matrices of Pascal-like Triangles Defined by Riordan Arrays</a>, arXiv:1805.02274 [math.CO], 2018.
%H A033282 Paul Barry, <a href="https://www.emis.de/journals/JIS/VOL22/Barry3/barry422.html">Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles</a>, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
%H A033282 Karin Baur and P. P. Martin, <a href="https://arxiv.org/abs/1601.05080">The fibres of the Scott map on polygon tilings are the flip equivalence classes</a>, arXiv:1601.05080 [math.CO], 2016.
%H A033282 David Beckwith, <a href="http://www.jstor.org/stable/2589081">Legendre polynomials and polygon dissections?</a>, Amer. Math. Monthly, 105 (1998), 256-257.
%H A033282 William Butler, Andrew Kalotay and N. J. A. Sloane, <a href="/A000108/a000108_3.pdf">Correspondence, 1974</a>
%H A033282 Arthur Cayley, <a href="http://plms.oxfordjournals.org/content/s1-22/1/237.extract">On the partitions of a polygon</a>, Proc. London Math. Soc., 22 (1891), 237-262 = Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 93ff. (See p. 239.)
%H A033282 Adrian Celestino and Yannic Vargas, <a href="https://arxiv.org/abs/2311.07824">Schröder trees, antipode formulas and non-commutative probability</a>, arXiv:2311.07824 [math.CO], 2023.
%H A033282 Frédéric Chapoton, <a href="http://www.mat.univie.ac.at/~slc/wpapers/s51chapoton.html">Enumerative properties of generalized associahedra</a>, Séminaire Lotharingien de Combinatoire, B51b (2004), 16 pp.
%H A033282 Johann Cigler, <a href="http://arxiv.org/abs/1501.04750">Some remarks and conjectures related to lattice paths in strips along the x-axis</a>, arXiv:1501.04750 [math.CO], 2015.
%H A033282 Manosij Ghosh Dastidar and Michael Wallner, <a href="https://arxiv.org/abs/2402.17849">Bijections and congruences involving lattice paths and integer compositions</a>, arXiv:2402.17849 [math.CO], 2024. See p. 16.
%H A033282 Jesús A. De Loera, Jörg Rambau, and Francisco Santos Leal, <a href="http://personales.unican.es/santosf/MSRI03/chapter1.pdf">Triangulations of Point Sets</a> [From Tom Copeland Oct 11 2011]
%H A033282 Satyan L. Devadoss, <a href="http://www.ams.org/notices/200406/fea-devadoss.pdf">Combinatorial Equivalence of Real Moduli Spaces</a>, Notices Amer. Math. Soc. 51 (2004), no. 6, 620-628.
%H A033282 Satyan L. Devadoss and Ronald C. Read, <a href="https://arxiv.org/abs/math/0008145">Cellular structures determined by polygons and trees</a>, arXiv/0008145 [math.CO], 2000. [From Tom Copeland Nov 21 2017]
%H A033282 Anton Dochtermann, <a href="http://arxiv.org/abs/1503.06243">Face rings of cycles, associahedra, and standard Young tableaux</a>, arXiv preprint arXiv:1503.06243 [math.CO], 2015.
%H A033282 Brian Drake, Ira M. Gessel, and Guoce Xin, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Gessel/gessel20.html">Three Proofs and a Generalization of the Goulden-Litsyn-Shevelev Conjecture on a Sequence Arising in Algebraic Geometry,</a> J. of Integer Sequences, Vol. 10 (2007), Article 07.3.7.
%H A033282 Cassandra Durell and Stefan Forcey, <a href="https://arxiv.org/abs/1905.09160">Level-1 Phylogenetic Networks and their Balanced Minimum Evolution Polytopes</a>, arXiv:1905.09160 [math.CO], 2019.
%H A033282 P. Flajolet and M. Noy, <a href="http://dx.doi.org/10.1016/S0012-365X(98)00372-0">Analytic Combinatorics of Non-crossing Configurations</a>, Discrete Math., 204, 1999, 203-229.
%H A033282 Sergey Fomin and Nathan Reading, <a href="https://arxiv.org/abs/math/0505518">Root systems and generalized associahedra</a>, Lecture notes for IAS/Park-City 2004, arXiv:math/0505518 [math.CO], 2005-2008. [From _Peter Bala_, Oct 28 2008]
%H A033282 Sergey Fomin and Andrei Zelevinsky, <a href="http://arXiv.org/abs/math/0104151">Cluster algebras I: Foundations</a>, arXiv:math/0104151 [math.RT], 2001.
%H A033282 Sergey Fomin and Andrei Zelevinsky, <a href="http://dx.doi.org/10.1090/S0894-0347-01-00385-X">Cluster algebras I: Foundations</a>, J. Amer. Math. Soc. 15 (2002) no.2, 497-529.
%H A033282 Sergey Fomin and Andrei Zelevinsky, <a href="http://www.jstor.org/stable/3597238">Y-systems and generalized associahedra</a>, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.
%H A033282 Ivan Geffner and Marc Noy, <a href="https://doi.org/10.37236/6249">Counting Outerplanar Maps</a>, Electronic Journal of Combinatorics 24(2) (2017), #P2.3.
%H A033282 Rijun Huang, Fei Teng, and Bo Feng, <a href="https://arxiv.org/abs/1801.08965">Permutation in the CHY-Formulation</a>, arXiv:1801.08965 [hep-th], 2018.
%H A033282 Germain Kreweras, <a href="http://dx.doi.org/10.1016/0012-365X(72)90041-6">Sur les partitions non croisées d'un cycle</a>, (French) Discrete Math. 1 (1972), no. 4, 333--350. MR0309747 (46 #8852).
%H A033282 Germain Kreweras, <a href="http://www.numdam.org/item?id=BURO_1973__20__3_0">Sur les hiérarchies de segments</a>, Cahiers Bureau Universitaire Recherche Opérationnelle, Cahier 20, Inst. Statistiques, Univ. Paris, 1973.
%H A033282 Germain Kreweras, <a href="/A001844/a001844.pdf">Sur les hiérarchies de segments</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
%H A033282 Germain Kreweras, <a href="http://archive.numdam.org/article/MSH_1976__53__5_0.pdf">Les préordres totaux compatibles avec un ordre partiel</a>, Math. Sci. Humaines No. 53 (1976), 5-30.
%H A033282 Germain Kreweras, <a href="/A019538/a019538.pdf">Les préordres totaux compatibles avec un ordre partiel</a>, Math. Sci. Humaines No. 53 (1976), 5-30. (Annotated scanned copy)
%H A033282 Thibault Manneville and Vincent Pilaud, <a href="http://arxiv.org/abs/1501.07152">Compatibility fans for graphical nested complexes</a>, arXiv:1501.07152 [math.CO], 2015.
%H A033282 Sebastian Mizera, <a href="https://doi.org/10.1007/JHEP08(2017)097">Combinatorics and topology of Kawai-Lewellen-Tye relations</a> J. High Energy Phys. 2017, No. 8, Paper No. 97, 54 p. (2017).
%H A033282 J.-C. Novelli and J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv:1403.5962 [math.CO], 2014.
%H A033282 Vincent Pilaud, <a href="http://arxiv.org/abs/1505.07665">Brick polytopes, lattice quotients, and Hopf algebras</a>, arXiv:1505.07665 [math.CO], 2015.
%H A033282 Vincent Pilaud and V. Pons, <a href="http://arxiv.org/abs/1606.09643">Permutrees</a>, arXiv:1606.09643 [math.CO], 2016-2017.
%H A033282 Ronald C. Read, <a href="http://dx.doi.org/10.1007/BF03031688">On general dissections of a polygon</a>, Aequat. Math. 18 (1978), 370-388.
%H A033282 Dean Rubine, <a href="https://arxiv.org/abs/2507.13045">Exercises for A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode</a>, arXiv:2507.13045 [math.CO], 2025. See p. 9.
%H A033282 Rodica Simion, <a href="http://dx.doi.org/10.1006/aama.1996.0505">Convex Polytopes and Enumeration</a>, Adv. in Appl. Math. 18 (1997) pp. 149-180.
%H A033282 Richard P. Stanley, <a href="http://dx.doi.org/10.1006/jcta.1996.0099">Polygon dissections and standard Young tableaux</a>, J. Comb. Theory, Ser. A, 76, 175-177, 1996.
%H A033282 Rekha R. Thomas, <a href="https://web.archive.org/web/20120615000000*/http://www.cis.upenn.edu/~cis610/ThomasLGC.pdf">Lectures in Geometric Combinatorics</a> [_Tom Copeland_, Oct 11 2011]
%F A033282 G.f. G = G(t, z) satisfies (1+t)*G^2 - z*(1-z-2*t*z)*G + t*z^4 = 0.
%F A033282 T(n, k) = binomial(n-3, k)*binomial(n+k-1, k)/(k+1) for n >= 3, 0 <= k <= n-3.
%F A033282 From _Tom Copeland_, Nov 03 2008: (Start)
%F A033282 Two g.f.s (f1 and f2) for A033282 and their inverses (x1 and x2) can be derived from the Drake and Barry references.
%F A033282 1. a: f1(x,t) = y = {1 - (2t+1) x - sqrt[1 - (2t+1) 2x + x^2]}/[2x (t+1)] = t x + (t + 2 t^2) x^2 + (t + 5 t^2 + 5 t^3) x^3 + ...
%F A033282 b: x1 = y/[t + (2t+1)y + (t+1)y^2] = y {1/[t/(t+1) + y] - 1/(1+y)} = (y/t) - (1+2t)(y/t)^2 + (1+ 3t + 3t^2)(y/t)^3 +...
%F A033282 2. a: f2(x,t) = y = {1 - x - sqrt[(1-x)^2 - 4xt]}/[2(t+1)] = (t/(t+1)) x + t x^2 + (t + 2 t^2) x^3 + (t + 5 t^2 + 5 t^3) x^4 + ...
%F A033282 b: x2 = y(t+1) [1- y(t+1)]/[t + y(t+1)] = (t+1) (y/t) - (t+1)^3 (y/t)^2 + (t+1)^4 (y/t)^3 + ...
%F A033282 c: y/x2(y,t) = [t/(t+1) + y] / [1- y(t+1)] = t/(t+1) + (1+t) y + (1+t)^2 y^2 + (1+t)^3 y^3 + ...
%F A033282 x2(y,t) can be used along with the Lagrange inversion for an o.g.f. (A133437) to generate A033282 and show that A133437 is a refinement of A033282, i.e., a refinement of the f-polynomials of the associahedra, the Stasheff polytopes.
%F A033282 y/x2(y,t) can be used along with the indirect Lagrange inversion (A134264) to generate A033282 and show that A134264 is a refinement of A001263, i.e., a refinement of the h-polynomials of the associahedra.
%F A033282 f1[x,t](t+1) gives a generator for A088617.
%F A033282 f1[xt,1/t](t+1) gives a generator for A060693, with inverse y/[1 + t + (2+t) y + y^2].
%F A033282 f1[x(t-1),1/(t-1)]t gives a generator for A001263, with inverse y/[t + (1+t) y + y^2].
%F A033282 The unsigned coefficients of x1(y t,t) are A074909, reverse rows of A135278. (End)
%F A033282 G.f.: 1/(1-x*y-(x+x*y)/(1-x*y/(1-(x+x*y)/(1-x*y/(1-(x+x*y)/(1-x*y/(1-.... (continued fraction). - _Paul Barry_, Feb 06 2009
%F A033282 Let h(t) = (1-t)^2/(1+(u-1)*(1-t)^2) = 1/(u + 2*t + 3*t^2 + 4*t^3 + ...), then a signed (n-1)-th row polynomial of A033282 is given by u^(2n-1)*(1/n!)*((h(t)*d/dt)^n) t, evaluated at t=0, with initial n=2. The power series expansion of h(t) is related to A181289 (cf. A086810). - _Tom Copeland_, Sep 06 2011
%F A033282 With a different offset, the row polynomials equal 1/(1 + x)*Integral_{0..x} R(n,t) dt, where R(n,t) = Sum_{k = 0..n} binomial(n,k)*binomial(n+k,k)*t^k are the row polynomials of A063007. - _Peter Bala_, Jun 23 2016
%F A033282 n-th row polynomial = ( LegendreP(n-1,2*x + 1) - LegendreP(n-3,2*x + 1) )/((4*n - 6)*x*(x + 1)), n >= 3. - _Peter Bala_, Feb 22 2017
%F A033282 n*T(n+1, k) = (4n-6)*T(n, k-1) + (2n-3)*T(n, k) - (n-3)*T(n-1, k) for n >= 4. - _Fang Lixing_, May 07 2019
%e A033282 The triangle T(n, k) begins:
%e A033282 n\k 0 1 2 3 4 5 6 7 8 9
%e A033282 3: 1
%e A033282 4: 1 2
%e A033282 5: 1 5 5
%e A033282 6: 1 9 21 14
%e A033282 7: 1 14 56 84 42
%e A033282 8: 1 20 120 300 330 132
%e A033282 9: 1 27 225 825 1485 1287 429
%e A033282 10: 1 35 385 1925 5005 7007 5005 1430
%e A033282 11: 1 44 616 4004 14014 28028 32032 19448 4862
%e A033282 12: 1 54 936 7644 34398 91728 148512 143208 75582 16796
%e A033282 ... reformatted. - _Wolfdieter Lang_, Mar 17 2017
%p A033282 T:=(n,k)->binomial(n-3,k)*binomial(n+k-1,k)/(k+1): seq(seq(T(n,k),k=0..n-3),n=3..12); # _Muniru A Asiru_, Nov 24 2018
%t A033282 t[n_, k_] = Binomial[n-3, k]*Binomial[n+k-1, k]/(k+1);
%t A033282 Flatten[Table[t[n, k], {n, 3, 12}, {k, 0, n-3}]][[1 ;; 52]] (* _Jean-François Alcover_, Jun 16 2011 *)
%o A033282 (PARI) Q=(1+z-(1-(4*w+2+O(w^20))*z+z^2+O(z^20))^(1/2))/(2*(1+w)*z);for(n=3,12,for(m=1,n-2,print1(polcoef(polcoef(Q,n-2,z),m,w),", "))) \\ _Hugo Pfoertner_, Nov 19 2018
%o A033282 (PARI) for(n=3,12, for(k=0,n-3, print1(binomial(n-3,k)*binomial(n+k-1,k)/(k+1), ", "))) \\ _G. C. Greubel_, Nov 19 2018
%o A033282 (Magma) [[Binomial(n-3, k)*Binomial(n+k-1, k)/(k+1): k in [0..(n-3)]]: n in [3..12]]; // _G. C. Greubel_, Nov 19 2018
%o A033282 (Sage) [[ binomial(n-3,k)*binomial(n+k-1,k)/(k+1) for k in (0..(n-3))] for n in (3..12)] # _G. C. Greubel_, Nov 19 2018
%Y A033282 Cf. diagonals: A000012, A000096, A033275, A033276, A033277, A033278, A033279; A000108, A002054, A002055, A002056, A007160, A033280, A033281; row sums: A001003 (Schroeder numbers, first term omitted). See A086810 for another version.
%Y A033282 A007160 is a diagonal. Cf. A001263.
%Y A033282 With leading zero: A086810.
%Y A033282 Cf. A019538 'faces' of the permutohedron.
%Y A033282 Cf. A063007 (f-vectors type B associahedra), A080721 (f-vectors type D associahedra), A126216 (mirror image).
%Y A033282 Cf. A248727 for a relation to f-polynomials of simplices.
%Y A033282 Cf. A111785 (contracted partition array, unsigned; see a comment above).
%Y A033282 Antidiagonal sums give A005043. - _Jordan Tirrell_, Jun 01 2017
%K A033282 nonn,tabl,easy
%O A033282 3,3
%A A033282 _N. J. A. Sloane_
%E A033282 Missing factor of 2 for expansions of f1 and f2 added by _Tom Copeland_, Apr 12 2009