A134685 -id:A134685 - cp's OEIS frontend

A033282 Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.

1, 1, 2, 1, 5, 5, 1, 9, 21, 14, 1, 14, 56, 84, 42, 1, 20, 120, 300, 330, 132, 1, 27, 225, 825, 1485, 1287, 429, 1, 35, 385, 1925, 5005, 7007, 5005, 1430, 1, 44, 616, 4004, 14014, 28028, 32032, 19448, 4862, 1, 54, 936, 7644, 34398, 91728, 148512, 143208, 75582, 16796

Offset: 3

N. J. A. Sloane

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
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).
Number of k-dimensional 'faces' of the n-dimensional associahedron (see Simion, p. 168). - Mitch Harris, Jan 16 2007
Mirror image of triangle A126216. - Philippe Deléham, Oct 19 2007
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
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
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
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
Diagonals of A132081 are rows of A033282. - Tom Copeland, May 08 2012
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
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
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
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
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

			The triangle T(n, k) begins:
n\k  0  1   2    3     4     5      6      7     8     9
3:   1
4:   1  2
5:   1  5   5
6:   1  9  21   14
7:   1 14  56   84    42
8:   1 20 120  300   330   132
9:   1 27 225  825  1485  1287    429
10:  1 35 385 1925  5005  7007   5005   1430
11:  1 44 616 4004 14014 28028  32032  19448  4862
12:  1 54 936 7644 34398 91728 148512 143208 75582 16796
... reformatted. - _Wolfdieter Lang_, Mar 17 2017

S. Devadoss and J. O'Rourke, Discrete and Computational Geometry, Princeton Univ. Press, 2011 (See p. 241.)
Ronald L. Graham, Donald E. Knuth, Oren Patashnik, Concrete Mathematics, 2nd ed., Addison-Wesley, 1994. Exercise 7.50, pages 379, 573.
T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Section 5.8.

Vincenzo Librandi, Table of n, a(n) for n = 3..2000
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009) 09.7.6.
Paul Barry, On the Inverses of a Family of Pascal-Like Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.6.
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Paul Barry, On the f-Matrices of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1805.02274 [math.CO], 2018.
Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
Karin Baur and P. P. Martin, The fibres of the Scott map on polygon tilings are the flip equivalence classes, arXiv:1601.05080 [math.CO], 2016.
David Beckwith, Legendre polynomials and polygon dissections?, Amer. Math. Monthly, 105 (1998), 256-257.
William Butler, Andrew Kalotay and N. J. A. Sloane, Correspondence, 1974
Arthur Cayley, On the partitions of a polygon, 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.)
Adrian Celestino and Yannic Vargas, Schröder trees, antipode formulas and non-commutative probability, arXiv:2311.07824 [math.CO], 2023.
Frédéric Chapoton, Enumerative properties of generalized associahedra, Séminaire Lotharingien de Combinatoire, B51b (2004), 16 pp.
Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015.
Manosij Ghosh Dastidar and Michael Wallner, Bijections and congruences involving lattice paths and integer compositions, arXiv:2402.17849 [math.CO], 2024. See p. 16.
Jesús A. De Loera, Jörg Rambau, and Francisco Santos Leal, Triangulations of Point Sets [From Tom Copeland Oct 11 2011]
Satyan L. Devadoss, Combinatorial Equivalence of Real Moduli Spaces, Notices Amer. Math. Soc. 51 (2004), no. 6, 620-628.
Satyan L. Devadoss and Ronald C. Read, Cellular structures determined by polygons and trees, arXiv/0008145 [math.CO], 2000. [From Tom Copeland Nov 21 2017]
Anton Dochtermann, Face rings of cycles, associahedra, and standard Young tableaux, arXiv preprint arXiv:1503.06243 [math.CO], 2015.
Brian Drake, Ira M. Gessel, and Guoce Xin, Three Proofs and a Generalization of the Goulden-Litsyn-Shevelev Conjecture on a Sequence Arising in Algebraic Geometry, J. of Integer Sequences, Vol. 10 (2007), Article 07.3.7.
Cassandra Durell and Stefan Forcey, Level-1 Phylogenetic Networks and their Balanced Minimum Evolution Polytopes, arXiv:1905.09160 [math.CO], 2019.
P. Flajolet and M. Noy, Analytic Combinatorics of Non-crossing Configurations, Discrete Math., 204, 1999, 203-229.
Sergey Fomin and Nathan Reading, Root systems and generalized associahedra, Lecture notes for IAS/Park-City 2004, arXiv:math/0505518 [math.CO], 2005-2008. [From _Peter Bala_, Oct 28 2008]
Sergey Fomin and Andrei Zelevinsky, Cluster algebras I: Foundations, arXiv:math/0104151 [math.RT], 2001.
Sergey Fomin and Andrei Zelevinsky, Cluster algebras I: Foundations, J. Amer. Math. Soc. 15 (2002) no.2, 497-529.
Sergey Fomin and Andrei Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.
Ivan Geffner and Marc Noy, Counting Outerplanar Maps, Electronic Journal of Combinatorics 24(2) (2017), #P2.3.
Rijun Huang, Fei Teng, and Bo Feng, Permutation in the CHY-Formulation, arXiv:1801.08965 [hep-th], 2018.
Germain Kreweras, Sur les partitions non croisées d'un cycle, (French) Discrete Math. 1 (1972), no. 4, 333--350. MR0309747 (46 #8852).
Germain Kreweras, Sur les hiérarchies de segments, Cahiers Bureau Universitaire Recherche Opérationnelle, Cahier 20, Inst. Statistiques, Univ. Paris, 1973.
Germain Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
Germain Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30.
Germain Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30. (Annotated scanned copy)
Thibault Manneville and Vincent Pilaud, Compatibility fans for graphical nested complexes, arXiv:1501.07152 [math.CO], 2015.
Sebastian Mizera, Combinatorics and topology of Kawai-Lewellen-Tye relations J. High Energy Phys. 2017, No. 8, Paper No. 97, 54 p. (2017).
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv:1403.5962 [math.CO], 2014.
Vincent Pilaud, Brick polytopes, lattice quotients, and Hopf algebras, arXiv:1505.07665 [math.CO], 2015.
Vincent Pilaud and V. Pons, Permutrees, arXiv:1606.09643 [math.CO], 2016-2017.
Ronald C. Read, On general dissections of a polygon, Aequat. Math. 18 (1978), 370-388.
Dean Rubine, Exercises for A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, arXiv:2507.13045 [math.CO], 2025. See p. 9.
Rodica Simion, Convex Polytopes and Enumeration, Adv. in Appl. Math. 18 (1997) pp. 149-180.
Richard P. Stanley, Polygon dissections and standard Young tableaux, J. Comb. Theory, Ser. A, 76, 175-177, 1996.
Rekha R. Thomas, Lectures in Geometric Combinatorics [_Tom Copeland_, Oct 11 2011]

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.
A007160 is a diagonal. Cf. A001263.
With leading zero: A086810.
Cf. A019538 'faces' of the permutohedron.
Cf. A063007 (f-vectors type B associahedra), A080721 (f-vectors type D associahedra), A126216 (mirror image).
Cf. A248727 for a relation to f-polynomials of simplices.
Cf. A111785 (contracted partition array, unsigned; see a comment above).
Antidiagonal sums give A005043. - Jordan Tirrell, Jun 01 2017

[[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

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[n_, k_] = Binomial[n-3, k]*Binomial[n+k-1, k]/(k+1);
Flatten[Table[t[n, k], {n, 3, 12}, {k, 0, n-3}]][[1 ;; 52]] (* Jean-François Alcover, Jun 16 2011 *)

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

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

[[ 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

G.f. G = G(t, z) satisfies (1+t)*G^2 - z*(1-z-2*t*z)*G + t*z^4 = 0.
T(n, k) = binomial(n-3, k)*binomial(n+k-1, k)/(k+1) for n >= 3, 0 <= k <= n-3.
From Tom Copeland, Nov 03 2008: (Start)
Two g.f.s (f1 and f2) for A033282 and their inverses (x1 and x2) can be derived from the Drake and Barry references.
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 + ...
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 +...
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 + ...
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 + ...
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 + ...
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.
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.
f1[x,t](t+1) gives a generator for A088617.
f1[xt,1/t](t+1) gives a generator for A060693, with inverse y/[1 + t + (2+t) y + y^2].
f1[x(t-1),1/(t-1)]t gives a generator for A001263, with inverse y/[t + (1+t) y + y^2].
The unsigned coefficients of x1(y t,t) are A074909, reverse rows of A135278. (End)
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
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
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
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
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

Missing factor of 2 for expansions of f1 and f2 added by Tom Copeland, Apr 12 2009

A134991 Triangle of Ward numbers T(n,k) read by rows.

Original entry on oeis.org

1, 1, 3, 1, 10, 15, 1, 25, 105, 105, 1, 56, 490, 1260, 945, 1, 119, 1918, 9450, 17325, 10395, 1, 246, 6825, 56980, 190575, 270270, 135135, 1, 501, 22935, 302995, 1636635, 4099095, 4729725, 2027025, 1, 1012, 74316, 1487200, 12122110, 47507460, 94594500, 91891800, 34459425

Offset: 1

Tom Copeland, Feb 05 2008

This is the triangle of associated Stirling numbers of the second kind, A008299, read along the diagonals.
This is also a row-reversed version of A181996 (with an additional leading 1) - see the table on p. 92 in the Ward reference. A134685 is a refinement of the Ward table.
The first and second diagonals are A001147 and A000457 and appear in the diagonals of several OEIS entries. The polynomials also appear in Carlitz (p. 85), Drake et al. (p. 8) and Smiley (p. 7).
First few polynomials (with a different offset) are
  P(0,t) = 0
  P(1,t) = 1
  P(2,t) = t
  P(3,t) = t + 3*t^2
  P(4,t) = t + 10*t^2 + 15*t^3
  P(5,t) = t + 25*t^2 + 105*t^3 + 105*t^4
These are the "face" numbers of the tropical Grassmannian G(2,n),related to phylogenetic trees (with offset 0 beginning with P(2,t)). Corresponding h-vectors are A008517. - Tom Copeland, Oct 03 2011
A133314 applied to the derivative of A(x,t) implies (a.+b.)^n = 0^n, for (b_n)=P(n+1,t) and (a_0)=1, (a_1)=-t, and (a_n)=-(1+t) P(n,t) otherwise. E.g., umbrally, (a.+b.)^2 = a_2*b_0 + 2 a_1*b_1 + a_0*b_2 = 0. - Tom Copeland, Oct 08 2011
Beginning with the second column, the rows give the faces of the Whitehouse simplicial complex with the fourth-order complex being three isolated vertices and the fifth-order being the Petersen graph with 10 vertices and 15 edges (cf. Readdy). - Tom Copeland, Oct 03 2014
Stratifications of smooth projective varieties which are fine moduli spaces for stable n-pointed rational curves. Cf. pages 20 and 30 of the Kock and Vainsencher reference and references in A134685. - Tom Copeland, May 18 2017
Named after the American mathematician Morgan Ward (1901-1963). - Amiram Eldar, Jun 26 2021

			Triangle begins:
  1
  1   3
  1  10   15
  1  25  105  105
  1  56  490 1260   945
  1 119 1918 9450 17325 10395
  ...

Louis Comtet, Advanced Combinatorics, Reidel, 1974, page 222.

G. C. Greubel, Rows n = 1..65, flattened
J. Fernando Barbero G., Jesús Salas and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences, I: General Structure, Journal of Combinatorial Theory, Series A, Vol. 125 (2014), pp. 146-165; arXiv preprint, arXiv:1307.2010 [math.CO], 2013-2014.
J. Fernando Barbero G., Jesús Salas and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. II. Applications, arXiv preprint arXiv:1307.5624 [math.CO], 2013.
J. Fernando Barbero G., Jesús Salas and Eduardo J. S. Villaseñor, Generalized Stirling permutations and forests: Higher-order Eulerian and Ward numbers, Electronic Journal of Combinatorics, Vol. 22, No. 3 (2015), #P3.37.
Andreas Blass, Natasha Dobrinen and Dilip Raghavan, The next best thing to a p-point, The Journal of Symbolic Logic, Vol. 80, No. 3 (2015), pp. 866-900; arXiv preprint, arXiv:1308.3790 [math.LO], 2013.
David Callan, Toufik Mansour, and Mark Shattuck Some identities for derangement and Ward number sequences and related bijections, Pure Mathematics and Applications, Vol. 25 (Edition 2), pp. 132-143, 2015.
L. Carlitz, The coefficients in an asymptotic expansion and certain related numbers, Duke Math. J., Vol. 35, No. 1 (1968), pp. 83-90.
Lane Clark, Asymptotic normality of the Ward numbers, Discrete Math. 203 (1999), no. 1-3, 41-48. [From _N. J. A. Sloane_, Feb 06 2012]
Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms.
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 3.
Bishal Deb and Alan D. Sokal, Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness, arXiv:2507.18959 [math.CO], 2025. See p. 6.
Satyan L. Devadoss, Daoji Huang and Dominic Spadacene, Polyhedral covers of tree space, SIAM Journal on Discrete Mathematics, Vol. 28, No. 3 (2014), pp. 1508-1514; arXiv preprint, arXiv:1311.0766 [math.CO], 2013.
S. Devadoss and J. Morava, Diagonalizing the genome I: navigation in tree spaces, arXiv:1009.3224v2 [math.AG], 2012.
S. Devadoss and O. Schuh, Cartography of Tree Space, Leonardo (MIT Press Direct), Vol. 53, Issue 3, 2019.
Andreas Dieckmann and Erik Vigren, A New Result in Form of Finite Triple Sums for a Series from Ramanujan's Notebooks, Symmetry (2022) Vol. 14, No. 6, 1090.
Ming-Jian Ding and Jiang Zeng, Proof of an explicit formula for a series from Ramanujan's Notebooks via tree functions, arXiv:2307.00566 [math.CO], 2023.
Brian Drake, Ira M. Gessel and Guoce Xin, Three Proofs and a Generalization of the Goulden-Litsyn-Shevelev Conjecture on a Sequence Arising in Algebraic Geometry, J. of Integer Sequences, Vol. 10 (2007), Article 07.3.7.
Joseph Felsenstein, The number of evolutionary trees, Systematic Biology, 27 (1978), pp. 27-33, 1978.
Giovanni Gaiffi, Nested sets, set partitions and Kirkman-Cayley dissection numbers, arXiv preprint arXiv:1404.3395 [math.CO], 2014.
David M. Jackson, Achim Kempf and Alejandro H. Morales, A robust generalization of the Legendre transform for QFT, Journal of Physics A: Mathematical and Theoretical, Vol. 50, No. 22 (2017), 225201; arXiv preprint, arXiv:1612.0046 [hep-th], 2017.
Bradley Robert Jones, On tree hook length formulae, Feynman rules and B-series, Master's thesis, Math Dept., Simon Fraser University, 2014. p. 23.
Joachim Kock and Israel Vainsencher, Kontsevich's Formula for Rational Plane Curves, Recife, 1999 (version 31/01/2003).
Toufik Mansour and Mark Shattuck, A polynomial generalization of some associated sequences related to set partitions, Periodica Mathematica Hungarica, Vol. 75, No. 2 (December 2017), pp. 398-412.
MathOverflow, Face numbers for tropical Grassmannian G'_2,7 simplicial complex?, 2011.
Andrew Elvey Price and Alan D. Sokal, Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials, arXiv:2001.01468 [math.CO], 2020.
Margaret A. Readdy, The pre-WDVV ring of physics and its topology, The Ramanujan Journal, Vol. 10, No. 2 (2005), pp. 269-281; preprint, 2002.
Lucas Randazzo, Arboretum for a generalisation of Ramanujan polynomials, The Ramanujan Journal, Vol. 54 (2019), pp. 1-14; arXiv preprint, arXiv:1905.02083 [math.CO], 2019.
Lucas Randazzo, Combinatoire bijective autour d'arbres et de chemins, doctoral thesis, Université Paris-Est, 2019.
Peter Regner, Phylogenetic Trees: Selected Combinatorial Problems, Master's Thesis, 2012, Institute of Discrete Mathematics and Geometry, TU Vienna, p. 52.
Fuquan Ren, Jean Yeh, and Roberta Rui Zhou, Context-Free Grammars for Several Triangular Arrays, Axioms, Vol. 11, Issue 6, 297, 2022.
E. G. Santos, Counting non-attacking chess pieces placements: Bishops and Anassas, arXiv:2411.16492 [math.CO], 2024. See p. 8.
Leonard M. Smiley, Completion of a Rational Function Sequence of Carlitz, arXiv:math/0006106 [math.CO], 2000.
Pierre Théorêt, Fonctions génératrices pour une classe d'équations aux différences partielles, Ann. Sci. Math. Quebec 19, 91-105 (1995).
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See p. 1.
Morgan Ward, The representations of Stirling's numbers and Stirling's polynomials as sums of factorials, Amer. J. Math., Vol. 56, No. 1 (1934), pp. 87-95.
Bao-Xuan Zhu, Coefficientwise Hankel-total positivity of row-generating polynomials for the m-Jacobi-Rogers triangle, arXiv:2202.03793 [math.CO], 2022.

The same as A269939, with column k = 0 removed.
A reshaped version of the triangle of associated Stirling numbers of the second kind, A008299.
A181996 is the mirror image.
Columns k = 2, 3, 4 are A000247, A000478, A058844.
Diagonal k = n is A001147.
Diagonal k = n - 1 is A000457.
Row sums are A000311.
Alternating row sums are signed factorials (-1)^(n-1)*A000142(n).
Cf. A112493.

t[n_, k_] := Sum[(-1)^i*Binomial[n, i]*Sum[(-1)^j*(k-i-j)^(n-i)/(j!*(k-i-j)!), {j, 0, k-i}], {i, 0, k}]; row[n_] := Table[t[k, k-n], {k, n+1, 2*n}]; Table[row[n], {n, 1, 9}] // Flatten (* Jean-François Alcover, Apr 23 2014, after A008299 *)

E.g.f. for the polynomials is A(x,t) = (x-t)/(t+1) + T{ (t/(t+1)) * exp[(x-t)/(t+1)] }, where T(x) is the Tree function, the e.g.f. of A000169. The compositional inverse in x (about x = 0) is B(x) = x + -t * [exp(x) - x - 1]. Special case t = 1 gives e.g.f. for A000311. These results are a special case of A134685 with u(x) = B(x).
From Tom Copeland, Oct 26 2008: (Start)
Umbral-Sheffer formalism gives, for m a positive integer and u = t/(t+1),
[P(.,t)+Q(.,x)]^m = [m Q(m-1,x) - t Q(m,x)]/(t+1) + sum(n>=1) { n^(n-1)[u exp(-u)]^n/n! [n/(t+1)+Q(.,x)]^m }, when the series is convergent for a sequence of functions Q(n,x).
Check: With t=1; Q(n,x)=0^n, for n>=0; and Q(-1,x)=0, then [P(.,1)+Q(.,x)]^m = P(m,1) = A000311(m).
(End)
Let h(x,t) = 1/(dB(x)/dx) = 1/(1-t*(exp(x)-1)), an e.g.f. in x for row polynomials in t of A019538, then the n-th row polynomial in t of the table A134991, P(n,t), is given by ((h(x,t)*d/dx)^n)x evaluated at x=0, i.e., A(x,t) = exp(x*P(.,t)) = exp(x*h(u,t)*d/du) u evaluated at u=0. Also, dA(x,t)/dx = h(A(x,t),t). - Tom Copeland, Sep 05 2011
The polynomials (1+t)/t*P(n,t) are the row polynomials of A112493. Let f(x) = (1+x)/(1-x*t). Then for n >= 0, P(n+1,t) is given by t/(1+t)*(f(x)*d/dx)^n(f(x)) evaluated at x = 0. - Peter Bala, Sep 30 2011
From Tom Copeland, Oct 04 2011: (Start)
T(n,k) = (k+1)*T(n-1,k) + (n+k+1)*T(n-1,k-1) with starting indices n=0 and k=0 beginning with P(2,t) (as suggested by a formula of David Speyer on MathOverflow).
T(n,k) = k*T(n-1,k) + (n+k-1)*T(n-1,k-1) with starting indices n=1 and k=1 of table (cf. Smiley above and Riordin ref.[10] therein).
P(n,t) = (1/(1+t))^n * Sum_{k>=1} k^(n+k-1)*(u*exp(-u))^k / k! with u=(t/(t+1)) for n>1; therefore, Sum_{k>=1} (-1)^k k^(n+k-1) x^k/k! = [1+LW(x)]^(-n) P{n,-LW(x)/[1+LW(x)]}, with LW(x) the Lambert W-Fct.
T(n,k) = Sum_{i=0..k} ((-1)^i binomial(n+k,i) Sum_{j=0..k-i} (-1)^j (k-i-j)^(n+k-i)/(j!(k-i-j)!)) from relation to A008299. (End)
The e.g.f. A(x,t) = -v * ( Sum_{j=>1} D(j-1,u) (-z)^j / j! ) where u = (x-t)/(1+t), v = 1+u, z = x/((1+t) v^2) and D(j-1,u) are the polynomials of A042977. dA/dx = 1/((1+t)(v-A)) = 1/(1-t*(exp(A)-1)). - Tom Copeland, Oct 06 2011
The general results on the convolution of the refined partition polynomials of A134685, 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
E.g.f.: C(u,t) = (u-t)/(1+t) - W( -((t*exp((u-t)/(1+t)))/(1+t)) ), where W is the principal value of the Lambert W-function. - Cheng Peng, Sep 11 2021
The function C(u,t) in the previous formula by Peng is precisely the function A(u,t) given in the initial 2008 formula of this section and the Oct 06 2011 formula from Copeland. As noted in A000169, Euler's tree function is T(x) = -LambertW(-x), where W(x) is the principal branch of Lambert's function, and T(x) is the e.g.f. of A000169. - Tom Copeland, May 13 2022

Reference to A181996 added by N. J. A. Sloane, Apr 05 2012

Further edits by N. J. A. Sloane, Jan 24 2020

A112486 Coefficient triangle for polynomials used for e.g.f.s for unsigned Stirling1 diagonals.

Original entry on oeis.org

1, 1, 1, 2, 5, 3, 6, 26, 35, 15, 24, 154, 340, 315, 105, 120, 1044, 3304, 4900, 3465, 945, 720, 8028, 33740, 70532, 78750, 45045, 10395, 5040, 69264, 367884, 1008980, 1571570, 1406790, 675675, 135135, 40320, 663696, 4302216, 14777620, 29957620

Offset: 0

Wolfdieter Lang, Sep 12 2005

The k-th diagonal of |A008275| appears as the k-th column in |A008276| with k-1 leading zeros.
The recurrence, given below, is derived from (d/dx)g1(k,x) - g1(k,x)= x*(d/dx)g1(k-1,x) + g1(k-1,x), k >= 1, with input g(-1,x):=0 and initial condition g1(k,0)=1, k >= 0. This differential recurrence for the e.g.f. g1(k,x) follows from the one for unsigned Stirling1 numbers.
The column sequences start with A000142 (factorials), A001705, A112487- A112491, for m=0,...,5.
The main diagonal gives (2*k-1)!! = A001147(k), k >= 1.
This computation was inspired by the Bender article (see links), where the Stirling polynomials are discussed.
The e.g.f. for the k-th diagonal, k >= 1, of the unsigned Stirling1 triangle |A008275| with k-1 leading zeros is g1(k-1,x) = exp(x)*Sum_{m=0..k-1} a(k,m)*(x^(k-1+m))/(k-1+m)!.
a(k,n) = number of lists with entries from [n] such that (i) each element of [n] occurs at least once and at most twice, (ii) for each i that occurs twice, all entries between the two occurrences of i are > i, and (iii) exactly k elements of [n] occur twice. Example: a(1,2)=5 counts 112, 121, 122, 211, 221, and a(2,2)=3 counts 1122,1221,2211. - David Callan, Nov 21 2011

			Triangle begins:
    1;
    1,    1;
    2,    5,     3;
    6,   26,    35,    15;
   24,  154,   340,   315,   105;
  120, 1044,  3304,  4900,  3465,   945;
  720, 8028, 33740, 70532, 78750, 45045, 10395;
k=3 column of |A008276| is [0,0,2,11,35,85,175,...] (see A000914), its e.g.f. exp(x)*(2*x^2/2! + 5* x^3/3! + 3*x^4/4!).

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7. - From _N. J. A. Sloane_, Feb 06 2013
C. M. Bender, D. C. Brody and B. K. Meister, Bernoulli-like polynomials associated with Stirling Numbers, arXiv:math-ph/0509008 [math-ph], 2005.
Wolfdieter Lang, First 10 rows.

Cf. A112007 (triangle for o.g.f.s for unsigned Stirling1 diagonals). A112487 (row sums).

A112486 := proc(n,k)
    if n < 0 or k<0 or  k> n then
        0 ;
    elif n = 0 then
        1 ;
    else
        (n+k)*procname(n-1,k)+(n+k-1)*procname(n-1,k-1) ;
    end if;
end proc: # R. J. Mathar, Dec 19 2013

A112486 [n_, k_] := A112486[n, k] = Which[n<0 || k<0 || k>n, 0, n == 0, 1, True, (n+k)*A112486[n-1, k]+(n+k-1)*A112486[n-1, k-1]]; Table[A112486[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 05 2014, after R. J. Mathar *)

a(k, m) = (k+m)*a(k-1, m) + (k+m-1)*a(k-1, m-1) for k >= m >= 0, a(0, 0)=1, a(k, -1):=0, a(k, m)=0 if k < m.
From Tom Copeland, Oct 05 2011: (Start)
With polynomials
P(0,t) = 0
P(1,t) = 1
P(2,t) = -(1 + t)
P(3,t) = 2 + 5 t + 3 t^2
P(4,t) = -( 6 + 26 t + 35 t^2 + 15 t^3)
P(5,t) = 24 + 154 t +340 t^2 + 315 t^3 + 105 t^4
Apparently, P(n,t) = (-1)^(n+1) PW[n,-(1+t)] where PW are the Ward polynomials A134991. If so, an e.g.f. for the polynomials is
  A(x,t) = -(x+t+1)/t - LW{-((t+1)/t) exp[-(x+t+1)/t]}, where LW(x) is a suitable branch of the Lambert W Fct. (e.g., see A135338). The comp. inverse in x (about x = 0) is B(x) = x + (t+1) [exp(x) - x - 1]. See A112487 for special case t = 1. These results are a special case of A134685 with u(x) = B(x), i.e., u_1=1 and (u_n)=(1+t) for n>0.
Let h(x,t) = 1/(dB(x)/dx) = 1/[1+(1+t)*(exp(x)-1)], an e.g.f. in x for row polynomials in t of signed A028246 , then P(n,t), is given by
(h(x,t)*d/dx)^n x, evaluated at x=0, i.e., A(x,t)=exp(x*h(u,t)*d/du) u, evaluated at u=0. Also, dA(x,t)/dx = h(A(x,t),t).
The e.g.f. A(x,t) = -v * Sum_{j>=1} D(j-1,u) (-z)^j / j! where u=-(x+t+1)/t, v=1+u, z=(1+t*v)/(t*v^2) and D(j-1,u) are the polynomials of A042977. dA/dx = -1/[t*(v-A)].(End)
A133314 applied to the derivative of A(x,t) implies (a.+b.)^n = 0^n, for (b_n)=P(n+1,t) and (a_0)=1, (a_1)=t+1, and (a_n)=t*P(n,t) otherwise. E.g., umbrally, (a.+b.)^2 = a_2*b_0 + 2 a_1*b_1 + a_0*b_2 =0. - Tom Copeland, Oct 08 2011
The row polynomials R(n,x) may be calculated using R(n,x) = 1/x^(n+1)*D^n(x), where D is the operator (x^2+x^3)*d/dx. - Peter Bala, Jul 23 2012
For n>0, Sum_{k=0..n} a(n,k)*(-1/(1+W(t)))^(n+k+1) = (t d/dt)^(n+1) W(t), where W(t) is Lambert W function. For t=-x, this gives Sum_{k>=1} k^(k+n)*x^k/k! = - Sum_{k=0..n} a(n,k)*(-1/(1+W(-x)))^(n+k+1). - Max Alekseyev, Nov 21 2019
Conjecture: row polynomials are R(n,x) = Sum_{i=0..n} Sum_{j=0..i} Sum_{k=0..j} (n+i)!*Stirling2(n+j-k,j-k)*x^k*(x+1)^(j-k)*(-1)^(n+j+k)/((n+j-k)!*(i-j)!*k!). - Mikhail Kurkov, Apr 21 2025

A112487 a(n) = Sum_{k=0..n} E2(n, k)*2^k, where E2(n, k) are the second-order Eulerian numbers A340556.

Original entry on oeis.org

1, 2, 10, 82, 938, 13778, 247210, 5240338, 128149802, 3551246162, 109979486890, 3764281873042, 141104799067178, 5749087305575378, 252969604725106090, 11955367835505775378, 603967991604199335722, 32479636694930586142802, 1852497140997527094395050

Offset: 0

Wolfdieter Lang, Sep 12 2005

Previous name: Row sums of triangle A112486.

Michael De Vlieger, Table of n, a(n) for n = 0..376
Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7. - From _N. J. A. Sloane_, Feb 06 2013
Kenny Barrese, Jennifer Elder, Pamela E. Harris, and Anthony Simpson, Enumerating Flat Fubini Rankings, arXiv:2504.06466 [math.CO], 2025. Conjecture 4.4. p. 14.
W. Steven Gray, Luis A. Duffaut Espinosa, and Kurusch Ebrahimi-Fard, Additive Networks of Chen-Fliess Series: Local Convergence and Relative Degree, arXiv:2104.08950 [eess.SY], 2021.
W. S. Gray and M. Thitsa, System Interconnections and Combinatorial Integer Sequences, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 March 2013, Digital Object Identifier: 10.1109/SSST.2013.6524939.
M. Thitsa and W. S. Gray, On the Radius of Convergence of Interconnected Analytic Nonlinear Input-Output Systems, SIAM Journal on Control and Optimization, Vol. 50, No. 5, pp. 2786-2813. - From _N. J. A. Sloane_, Dec 26 2012

Cf. A340556, A112486, A032188, A032034, A134991, A135338.

A112487 := proc(n)
    add(A112486(n,k),k=0..n) ;
end proc: # R. J. Mathar, Dec 19 2013
seq(op(k, convert(asympt(GAMMA(n, 2*n)*exp(2*n)/(2*n)^n, n, 20), polynom))*(-1)^(k+1)*n^k, k = 1..19); # Maple 2017, Vaclav Kotesovec, Aug 14 2017
E2 := (n, k) -> `if`(k=0, k^n, combinat:-eulerian2(n, k-1));
a := n -> add(E2(n, k)*2^k, k=0..n):
seq(a(n), n=0..17); # Peter Luschny, Feb 13 2021

a[n_] := (n-1)!*(Sum[ Binomial[n+k-1, n-1]* Sum[(-1)^(n+j-1)*Binomial[k, j]* Sum[(Binomial[j, l]*(j-l)!*2^(j-l)*(-1)^l*StirlingS2[n-l+j-1, j-l])/(n-l+j-1)!, {l, 0, j}], {j, 0, k}], {k, 0, n-1}]); Table[a[n], {n, 1, 18}] (* Jean-François Alcover, Feb 26 2013, after Vladimir Kruchinin *)
T[n_, k_] := T[n, k] = If[k == 0, Boole[n == 0], If[n < 0, 0, k T[n - 1, k] + (2 n - k) T[n - 1, k - 1]]]; a[n_] := Sum[T[n, k] 2^k, {k, 0, n}];
Table[a[n], {n, 0, 17}] (* Peter Luschny, Feb 13 2021 *)

a(n):=n!*(sum(binomial(n+k, n)*sum((-1)^(n+j)*binomial(k, j)*sum((binomial(j, l)*(j-l)!*2^(j-l)*(-1)^l*stirling2(n-l+j, j-l))/(n-l+j)!, l, 0, j), j, 0, k), k, 0, n)); /* Vladimir Kruchinin, Feb 14 2012 */

{a(n)=local(A=1+x);for(i=1,n,A=exp(intformal(A+A^2)+x*O(x^n)));n!*polcoeff(A,n)} \\ Paul D. Hanna, Jun 30 2009

a(n) = Sum_{m=0..n} A112486(n, m), n >= 0.
a(n) = 2*A032188(n+1), n > 0. - Vladeta Jovovic, Jul 11 2007
From Paul D. Hanna, Jun 30 2009: (Start)
E.g.f. A(x) satisfies: A'(x) = A(x)^2 + A(x)^3.
E.g.f. A(x) satisfies: A(x) = exp( Integral[A(x) + A(x)^2]dx ) with A(0)=1. (End)
E.g.f. A(x) satisfies: A(x) = 2*exp(A(x)) - (2+x), where A(x) = Sum_{n>=0} a(n)*x^(n+1)/(n+1)! (the e.g.f. when offset=1). - Paul D. Hanna, Sep 23 2011
From Tom Copeland, Oct 05 2011: (Start)
With c(0)= 0 and c(n+1)= (-1)^n a(n) for n>=0, c(n)=(-1)^(n+1) PW(n,-2) with PW the Ward polynomials A134991. E.g.f. for the c(n) is A(x) = -(x+2)-LW{-2 exp[-(x+2)]}, where LW(x) is a suitable branch of the Lambert W Fct. (see A135338).
The compositional inverse is B(x) = x + 2(exp(x) - x - 1). These results are a special case of A134685 with u(x)=B(x), i.e., u_1=1 and (u_n)=2 for n>0.
Let h(x) = 1/(dB(x)/dx) = 1/[1+2(exp(x)-1)], then c(n) is given by (h(x)*d/dx)^n x, evaluated at x=0, i.e., A(x) = exp(x*h(u)*d/du) u, evaluated at u=0. Also, dA(x)/dx = h(A(x)).
The e.g.f. A(x) = -v * Sum_(j>=1) D(j-1,u) (-z)^j/ j! where u=-(x+2), v=1+u, z=(1+v)/(v^2) and D(j-1,u) are the polynomials of A042977. (End)
a(n) = n!*Sum_{k=0..n} binomial(n+k, n)*Sum_{j=0..k} (-1)^(n+j)*binomial(k, j)*Sum_{l=0..j} binomial(j, l)*(j-l)!*2^(j-l)*(-1)^l*Stirling2(n-l+j, j-l)/(n-l+j)!. - Vladimir Kruchinin, Feb 14 2012
G.f.: 1/Q(0), where Q(k)= 1 + k*x - 2*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 01 2013
a(n) ~ n^n / (exp(n) * (1-log(2))^(n+1/2)). - Vaclav Kotesovec, Aug 14 2017
a(0) = 1; a(n) = n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * a(k) * a(n-k-1). - Ilya Gutkovskiy, Jul 02 2020

New name from Peter Luschny, Feb 13 2021

A074060 Graded dimension of the cohomology ring of the moduli space of n-pointed stable curves of genus 0 satisfying the associativity equations of physics (also known as the WDVV equations).

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 16, 16, 1, 1, 42, 127, 42, 1, 1, 99, 715, 715, 99, 1, 1, 219, 3292, 7723, 3292, 219, 1, 1, 466, 13333, 63173, 63173, 13333, 466, 1, 1, 968, 49556, 429594, 861235, 429594, 49556, 968, 1, 1, 1981, 173570, 2567940, 9300303, 9300303, 2567940, 173570, 1981, 1

Offset: 3

Margaret A. Readdy, Aug 16 2002

Combinatorial interpretations of Lagrange inversion (A134685) and the 2-Stirling numbers of the first kind (A049444 and A143491) provide a combinatorial construction for A074060 (see first Copeland link). For relations of A074060 to other arrays see second Copeland link page 19. - Tom Copeland, Sep 28 2008

These Poincare polynomials for the compactified moduli space of rational curves are presented on p. 5 of Lando and Zvonkin as well as those for the non-compactified Poincare polynomials of A049444 in factorial form. - Tom Copeland, Jun 13 2021

			Viewed as a triangular array, the values are
  1;
  1,   1;
  1,   5,   1;
  1,  16,  16,   1;
  1,  42, 127,  42,   1; ...

Tom Copeland, Combinatorics of OEIS-A074060, Posted Sept. 2008.
Tom Copeland, Mathemagical Forests v2, Posted June 2008.
S. Keel, Intersection theory of moduli space of stable n-pointed curves of genus zero, Trans. Amer. Math. Soc. 330 (1992), 545-574.
M. Kontsevich and Y. Manin, Quantum cohomology of a product, (with Appendix by R. Kaufmann), Inv. Math. 124, f. 1-3 (1996) 313-339.
M. Kontsevich and Y. Manin, Quantum cohomology of a product, arXiv:q-alg/9502009, 1995.
S. Lando and A. Zvonkin, Graphs on surfaces and their applications, Encyclopaedia of Mathematical Sciences, 141, Springer, 2004.
Y. Manin, Generating functions in algebraic geometry and sums over trees, arXiv:alg-geom/9407005, 1994. - _Tom Copeland_, Dec 10 2011
M. A. Readdy, The pre-WDVV ring of physics and its topology, preprint, 2002.

Cf. A074059. 2nd diagonal is A002662.

DA:=((1+t)*A(u,t)+u)/(1-t*A(u,t)): F:=0: for k from 1 to 10 do F:=map(simplify,int(series(subs(A(u,t)=F,DA),u,k),u)); od: # Eric Rains, Apr 02 2005

DA = ((1+t) A[u, t] + u)/(1 - t A[u, t]); F = 0;
Do[F = Integrate[Series[DA /. A[u, t] -> F, {u, 0, k}], u], {k, 1, 10}];
(cc = CoefficientList[#, t]; cc Denominator[cc[[1]]])& /@ Drop[ CoefficientList[F, u], 2] // Flatten (* Jean-François Alcover, Oct 15 2019, after Eric Rains *)

Define offset to be 0 and P(n,t) = (-1)^n Sum_{j=0..n-2} a(n-2,j)*t^j with P(1,t) = -1 and P(0,t) = 1, then H(x,t) = -1 + exp(P(.,t)*x) is the compositional inverse in x about 0 of G(x,t) in A049444. H(x,0) = exp(-x) - 1, H(x,1) = -1 + exp( 2 + W( -exp(-2) * (2-x) ) ) and H(x,2) = 1 - (1+2*x)^(1/2), where W is a branch of the Lambert function such that W(-2*exp(-2)) = -2. - Tom Copeland, Feb 17 2008
Let offset=0 and g(x,t) = (1-t)/((1+x)^(t-1)-t), then the n-th row polynomial of the table is given by [(g(x,t)*D_x)^(n+1)]x with the derivative evaluated at x=0. - Tom Copeland, Jun 01 2008
With the notation in Copeland's comments, dH(x,t)/dx = -g(H(x,t),t). - Tom Copeland, Sep 01 2011
The term linear in x of [x*g(d/dx,t)]^n 1 gives the n-th row polynomial with offset 1. (See A134685.) - Tom Copeland, Oct 21 2011

More terms from Eric Rains, Apr 02 2005

A356145 Coefficients of the inverse refined Eulerian partition polynomials [E]^{-1}, partitional inverse to A145271. Irregular triangle read by row with lengths A000041.

Original entry on oeis.org

1, 1, -1, 1, 3, -4, 1, -15, 25, -4, -7, 1, 105, -210, 70, 60, -15, -11, 1, -945, 2205, -1120, -630, 70, 350, 126, -15, -26, -16, 1, 10395, -27720, 18900, 7875, -2800, -6930, -1638, 560, 455, 784, 238, -56, -42, -22, 1, -135135, 405405, -346500, -114345, 84700

Offset: 0

Tom Copeland, Jul 27 2022

These are the coefficients of the inverse refined Eulerian partitions polynomials, the substitutional inverse to the refined Eulerian partition polynomials [E] of A145271. [E] and [E]^{-1} are a conjugate dual with respect to the permutahedra polynomials [P] of A133314 (see formula section).

			The first few rows of coefficients with monomials in reverse order to the partitions of Abramowitz and Stegun (link in A000041, pp. 831-2) are
0)       1;
1)       1;
2)      -1,      1;
3)       3,     -4,       1;
4)     -15,     25,      -4,      -7,     1;
5)     105,   -210,      70,      60,   -15,    -11,     1;
6)    -945,   2205,   -1120,    -630,    70,    350,   126,   -15,    -26,    -16,      1;
7)   10395, -27720,   18900,    7875, -2800,  -6930, -1638,   560,    455,    784,    238,   -56,  -42,  -22,    1;
8) -135135, 405405, -346500, -114345, 84700, 138600, 24255, -2800, -27300, -11025, -18900, -3780, 1575, 1344, 2142, 1596, 414, -56, -98, -64, -29, 1;
    ...
The first few partition polynomials are
E_0^{(-1)} = 1,
E_1^{(-1)} = a1,
E_2^{(-1)} = -a1^2 + a2,
E_3^{(-1)} = 3 a1^3 - 4 a1 a2 + a3,
E_4^{(-1)} = -15 a1^4 + 25 a1^2 a2 - 4 a2^2 - 7 a1 a3 + a4,
E_5^{(-1)} = 105 a1^5 - 210 a1^3 a2 + 70 a1 a2^2 + 60 a1^2 a3 - 15 a2 a3 - 11 a1 a4 + a5,
E_6^{(-1)} = -945 a1^6 + 2205 a1^4 a2 - 1120 a1^2 a2^2 - 630 a1^3 a3 + 70 a2^3 + 350 a1 a2 a3 + 126 a1^2 a4 - 15 a3^2 - 26 a2 a4 - 16 a1 a5 + a6,
E_7^{(-1)} = 10395 a1^7 - 27720 a1^5 a2 + 18900 a1^3 a2^2 + 7875 a1^4 a3 - 2800 a1 a2^3 - 6930 a1^2 a2 a3 - 1638 a1^3 a4 + 560 a2^2 a3 + 455 a1 a3^2 + 784 a1 a2 a4 + 238 a1^2 a5 - 56 a3 a4 - 42 a2 a5 - 22 a1 a6 + a7,
E_8^{(-1)} = -135135 a1^8 + 405405 a1^6 a2 - 346500 a1^4 a2^2 - 114345 a1^5 a3 + 84700 a1^2 a2^3 + 138600 a1^3 a2 a3 + 24255 a1^4 a4 - 2800 a2^4 - 27300 a1 a2^2 a3 - 11025 a1^2 a3^2 - 18900 a1^2 a2 a4 - 3780 a1^3 a5 + 1575 a2 a3^2 + 1344 a2^2 a4 + 2142 a1 a3 a4 + 1596 a1 a2 a5 + 414 a1^2 a6 - 56 a4^2 - 98 a3 a5 - 64 a2 a6 - 29 a1ma7 + a8,
... .
Example substitution identities:
With the permutahedra polynomials
P_1 = -a_1,
P_2 = 2*a_1^2 - a_2,
P_3 = -6*a_1^3 + 6*a_2*a_1 - a_3,
the refined Eulerian polynomials
E_1 = a_1,
E_2 = a_1^2 + a_2,
E_3 = a_1^3 + 4*a_1*a_2 + a_3,
the reciprocal tangent polynomials
RT_1 = -a_1,
RT_2 = -a_2 + a_1^2,
RT_3 = -a_3 + 2*a_1*a_2 - a_1^3,
the Lagrange inversion polynomials
L_1 = -a_1,
L_2 = 3*a_1^2 - a_2,
L_3 = -15*a_1^3 + 10*a_1a_2 - a_3,
then
E^{-1}_3 = P_3(L_1,L_2,L_3) = -6*(-a_1)^3 + 6*(3*a_1^2 - a_2)*(-a_1) - (-15*a_1^3 + 10*a_1*a_2 - a_3) = 3*a_1^3 - 4*a_2*a_1 + a_3,
E^{-1}_3 = RT_3(P_1,P_2,P_3) = -(-6*a_1^3 + 6*a_2*a_1 - a_3) + 2*(-a_1)*(2*a_1^2 - a_2) - (-a_1)^3 = 3*a_1^3 - 4*a_2*a_1 + a_3,
E{-1}_3(E_1,E_2,E_3) = 3*a_1^3 - 4*a_1*(a_1^2 + a_2) + (a_1^3 + 4*a_1*a_2 + a_3) = a_3.

Tom Copeland, Matryoshka Dolls: Iterated noncrossing partitions, the refined Narayana group, and quantum fields, 2022.
Tom Copeland, One Matrix to Rule Them All, 2022.
Tom Copeland, The reduced inverse refined Eulerian polynomials and associated arrays, 2022.

Cf. A000041, A006351, A112493, A124324, A133314, A134685, A134991, A145271, A356144.

rows[nn_] := {{1}}~Join~With[{s = 1/D[InverseSeries[x + Sum[c[k - 1] x^k/k!, {k, 2, nn}] + O[x]^(nn + 1)], x]}, Table[Coefficient[n! s, x^n Product[c[t], {t, p}]], {n, nn-1}, {p, Reverse[Sort[Sort /@ IntegerPartitions[n]]]}]];
rows[8] // Flatten (* Andrey Zabolotskiy, Feb 17 2024 *)

B. = PolynomialRing(ZZ)
A. = PowerSeriesRing(B)
f =  x + a1*x^2/factorial(2) + a2*x^3/factorial(3) + a3*x^4/factorial(4) + a4*x^5/factorial(5) + a5*x^6/factorial(6) + a6*x^7/factorial(7) + a7*x^8/factorial(8) + a8*x^9/factorial(9) + a9*x^10/factorial(10)
g = f.reverse()
w = derivative(g,x)
I = 1 / w
# Added by Peter Luschny, Feb 17 2024:
for n, c in enumerate(I.list()[:9]):
    print(f"E[{n}]", (factorial(n)*c).coefficients())

Given the formal Taylor series or e.g.f. f(x) = x + a_1 x^2/2! + a_2 x^3/3! + ...,
E_n^{-1}(a_1,a_2,...,a_n) = D_{x=0}^n 1 / (D_x f^{(-1)}(x)), where D_x is the derivative w.r.t. x and f^{(-1)}(x) is the (possibly formal) compositional inverse of f(x) about the origin.
E_n^{-1}(a_1,a_2,...,a_n) = D_{x=0}^n 1 f'(f^{(-1)}(x)) by the inverse function theorem, where the prime indicates differentiation w.r.t. the argument of the function f. Note the correspondence to the analytic definitions of the reciprocal tangents [RT] of A356144, consistent with the following algebraic identities.
[E]^{-1} = [P][L] = [P][E][P] = [RT][P], representing, e.g., the substitution of the permutahedra polynomials [P] of A133314 for the indeterminates of the reciprocal tangent polynomials [RT] of A356144. [E] are the refined Eulerian polynomials of A145271, and [L], the classic Lagrange inversion polynomials of A134685.
Since [P]^2 = [L]^2 = [RT]^2 = [I], the substitutional identity, i.e., [P], [L], and [RT] are involutive transformations, many identities follow from the basic ones above, e.g., [L] = [P][E]^{-1} gives an inversion formula for a formal e.g.f. f(x) = x + a_1 x^2/2! + a_2 x^3/3! + ..., and we can identify [E] and [E]^{-1} as a conjugate dual.
With a_n = -x, [E]^{-1} reduces to a signed version of A112493 with an additional initial row, with the row sums of the unsigned coefficients being (1, A006351). A112493 is also given by the diagonals of A124324. See my link above on the reduced polynomials and associated arrays for more detail.
The sequence of row sums of the signed coefficients, i.e., E^{-1}(1,1,...,1), is the sequence (1, 1, 0, 0, 0, 0, ...).
Conjecture: row polynomials are R(n,1) for n > 0 where R(n,k) = R(n-1,k+1) - Sum_{j=1..n-1} binomial(n-1,j-1)*R(j,k)*R(n-j,1) for n > 1, k > 0 with R(1,k) = a_k for k > 0. - Mikhail Kurkov, Mar 22 2025

A119274 Triangle of coefficients of numerators in Padé approximation to exp(x).

Original entry on oeis.org

1, 2, 1, 12, 6, 1, 120, 60, 12, 1, 1680, 840, 180, 20, 1, 30240, 15120, 3360, 420, 30, 1, 665280, 332640, 75600, 10080, 840, 42, 1, 17297280, 8648640, 1995840, 277200, 25200, 1512, 56, 1, 518918400, 259459200, 60540480, 8648640, 831600, 55440, 2520

Offset: 0

Paul Barry, May 12 2006

n-th numerator of Padé approximation is (1/n!)*sum{j=0..n, C(n,j)(2n-j)!x^j}. Reversal of A113025. Row sums are A001517. First column is A001813. Inverse is A119275.
Also the Bell transform of the quadruple factorial numbers Product_{k=0..n-1} (4*k+2) (A001813) adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428 and for cross-references A265606. - Peter Luschny, Dec 31 2015
Dividing each diagonal by its initial element generates A054142. - Tom Copeland, Oct 10 2016

			Triangle begins
1,
2, 1,
12, 6, 1,
120, 60, 12, 1,
1680, 840, 180, 20, 1,
30240, 15120, 3360, 420, 30, 1

Cf. A001497, A054142.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> (2*n)!/n!, 9); # Peter Luschny, Jan 27 2016

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[(2#)!/#!&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)

# uses[bell_transform from A264428]
# Adds a column 1,0,0,0,... at the left side of the triangle.
def A119274_row(n):
    multifact_4_2 = lambda n: prod(4*k + 2 for k in (0..n-1))
    mfact = [multifact_4_2(k) for k in (0..n)]
    return bell_transform(n, mfact)
[A119274_row(n) for n in (0..9)] # Peter Luschny, Dec 31 2015

Number triangle T(n,k) = C(n,k)(2n-k)!/n!.

After adding a leading column (1,0,0,0,...), the triangle gives the coefficients of the Sheffer associated sequence (binomial-type polynomials) for the delta (lowering) operator D(1-D) with e.g.f. exp[ x * (1 - sqrt(1-4t)) / 2 ] . See Mathworld on Sheffer sequences. See A134685 for relation to Catalan numbers. - Tom Copeland, Feb 09 2008

A268442 Triangle read by rows, the coefficients of the inverse Bell polynomials.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, 3, -1, -3, 1, 0, -15, 10, -1, 15, -4, -6, 1, 0, 105, -105, 10, 15, -1, -105, 60, -5, 45, -10, -10, 1, 0, -945, 1260, -280, -210, 35, 21, -1, 945, -840, 70, 105, -6, -420, 210, -15, 105, -20, -15, 1

Offset: 0

Peter Luschny, Feb 06 2016

The triangle of coefficients of the Bell polynomials is A268441. For the definition of the inverse Bell polynomials see the link 'Bell transform'.

			[[1]],
[[0], [1]],
[[0], [-1],                    [1]],
[[0], [3, -1],                 [-3],           [1]],
[[0], [-15, 10, -1],           [15, -4],       [-6],      [1]],
[[0], [105, -105, 10, 15, -1], [-105, 60, -5], [45, -10], [-10], [1]]
Replacing the sublists by their sums reduces the triangle to the triangle of the Stirling numbers of first kind (A048994). The column 1 of sublists is A176740 (missing the leading 1) and A134685 in different order.

Peter Luschny, First 21 rows, flattened
Peter Luschny, The Bell transform
Peter Luschny, SageMath implementation

Cf. A036040, A048994, A134685, A176740, A268441.

A268442Matrix[dim_] := Module[ {v, r, A},
v = Table[Subscript[x,j],{j,1,dim}];
r = Table[Subscript[x,j]->1,{j,1,n}];
A = Table[Table[BellY[n,k,v], {k,0,dim}], {n,0,dim}];
Table[Table[MonomialList[Inverse[A][[n,k]]/. r[[1]],
v, Lexicographic] /. r, {k,1,n}] // Flatten, {n,1,dim}]];
A268442Matrix[7] // Flatten

Sage
```
# see link
```

A135338 Triangle read by rows: row n gives coefficients C(n,j) for a Sheffer sequence (binomial-type) with raising operator -x { 1 + W[ -exp(-2) * (2+D) ] } where W is the Lambert W multi-valued function.

Original entry on oeis.org

1, -1, 1, 1, -3, 1, -2, 7, -6, 1, 6, -20, 25, -10, 1, -24, 76, -105, 65, -15, 1, 120, -364, 511, -385, 140, -21, 1, -720, 2108, -2940, 2401, -1120, 266, -28, 1, 5040, -14328, 19720, -16632, 8841, -2772, 462, -36, 1, -40320, 111816, -151620, 129340, -73605, 27237, -6090, 750, -45, 1

Offset: 1

Tom Copeland, Feb 15 2008

The lowering (or delta) operator for these polynomials is L = -1 + exp{ 2 + W[ -exp(-2) * (2+D) ] } = Sum_{j >= 1} A074059(j) * D^j / j!.
The raising operator is R = -x { 1 + W[ -exp(-2) * (2+D) ] } = x { 1 + Sum_{j >= 1} (-1)^j * PW(j-1,-2) * D^j / j! }, where PW(j-1,x) are the polynomials of A042977.
W(x) here is W_-1 in the Monir reference and, about x = 0, W[ -exp(-2) * (2+x) ] = -[ 2 + Sum_{j >= 1} (-1)^j * PW(j-1,-2) * x^j / j! ].
From the relation between the delta and raising operators for associated binomial-type polynomials, A074059 = (1,1,2,7,34,...) and S = (1,-PW(0,-2),PW(1,-2),-PW(2,-2),...) = (1, -1, 0, -1, -2, -13, -74, -593, -5298, ...) form a list partition transform pair (see A133314); i.e., S and A074059 have reciprocal e.g.f.s and satisfy mutual recursion relations. Applying Faa di Bruno's formula to L gives other interesting integer relations between S and A074059.
The Bell transform of (-1)^n*factorial(n-1) if n>0, else 1. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

			Triangle read by rows:
     1;
    -1,    1;
     1,   -3,     1;
    -2,    7,    -6,    1;
     6,  -20,    25,  -10,     1;
   -24,   76,  -105,   65,   -15,   1;
   120, -364,   511, -385,   140, -21,   1;
  -720, 2108, -2940, 2401, -1120, 266, -28, 1;
...
From _R. J. Mathar_, Mar 22 2013: (Start)
The matrix inverse starts:
     1;
     1,    1;
     2,    3,    1;
     7,   11,    6,   1;
    34,   55,   35,  10,   1;
   213,  349,  240,  85,  15,  1;
  1630, 2695, 1939, 770, 175, 21, 1;
  ... (End)

Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).
F. Chapeau-Blondeau and A. Monir, Numerical Evaluation of the Lambert W Function and Application to Generation of Generalized Gaussian Noise With Exponent 1/2, IEEE Trans. on Signal Processing, Vol. 50, No. 9, Sept. 2002, p. 2160-2164.

Cf. A134685, A135494.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> `if`(n=0,1,(-1)^n*(n-1)!), 9); # Peter Luschny, Jan 27 2016

max = 10; s = Series[Exp[t*(2*x-(1+x)*Log[1+x])], {x, 0, max}, {t, 0, max}] // Normal; c[n_, j_] := SeriesCoefficient[s, {x, 0, n}, {t, 0, j}]*n!; Table[c[n, j], {n, 1, max}, {j, 1, n}] // Flatten (* Jean-François Alcover, Apr 23 2014, after Peter Bala, duplicate of Copeland's e.g.f. *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[Function[n, If[n == 0, 1, (-1)^n (n-1)!]], rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 26 2018, after Peter Luschny *)

# uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(lambda n: (-1)^n*factorial(n-1) if n>0 else 1, 10) # Peter Luschny, Jan 18 2016

The row polynomials P(n,t) = Sum_{j=1..n} C(n,j) * t^j satisfy exp[P(.,t) * x] = exp{ -t * [(1+x) * log(1+x) - 2*x] }, with P(0,t) = 1 and [ P(.,x) + P(.,y) ]^n = P(n,x+y). Here, as in the e.g.f., the umbral maneuver P(.,t)^n = P(n,t) is assumed. See Mathworld and Wikipedia on Sheffer sequences and umbral calculus for other general formulas, including expansion theorems.
From Peter Bala, Dec 09 2011: (Start)
E.g.f.: exp(t*(2*x-(1+x)*log(1+x))) = 1 + t*x + (t^2-t)*x^2/2! + (t^3-3*t^2+t)*x^3/3! + ... (Restatement of Copeland's e.g.f. above in umbral notation with P(.,t)^n = P(n,t).).
If a triangular array has an e.g.f. of the form exp(t*F(x)) with F(0) = 0, then the o.g.f.'s for the diagonals of the triangle are rational functions in t (see the Bala link). The rational functions are the coefficients in the compositional inverse (with respect to x) (x-t*F(x))^(-1). In this case (x-t*(2*x-(1+x)*log(1+x)))^(-1) = x/(1-t) - t/(1-t)^3*x^2/2! + (t+2*t^2)/(1-t)^5*x^3/3! - (2*t+6*t^2+7*t^3)/(1-t)^7*x^4/4! + ... . So, for example, the (unsigned) third subdiagonal has o.g.f. (2*t+6*t^2+7*t^3)/(1-t)^7 = 2*t + 20*t^2 + 105*t^3 + 385*t^4 + ... .
(End)

More terms from Jean-François Alcover, Apr 23 2014

A176740 Inversion of e.g.f. formal power series. Partition array in Abramowitz-Stegun (A-St) order.

Original entry on oeis.org

-1, -1, 3, -1, 10, -15, -1, 15, 10, -105, 105, -1, 21, 35, -210, -280, 1260, -945, -1, 28, 56, 35, -378, -1260, -280, 3150, 6300, -17325, 10395, -1, 36, 84, 126, -630, -2520, -1575, -2100, 6930, 34650, 15400, -51975, -138600, 270270, -135135, -1, 45, 120, 210, 126, -990, -4620, -6930, -4620, -5775

Offset: 0

Wolfdieter Lang, Jul 14 2010

Compare with A134685 which uses a different order with fewer entries.
For the inversion (aka reversion) of o.g.f. formal power series see A111785, and also A133437.
The sequence of row lengths of this array is p(n)=A000041(n) (number of partitions of n).
The unsigned triangle, with entries for like parts number m summed, is A134991 (2-associated Stirling numers of the second kind).
The row sums are A133942(n) = ((-1)^n) * n!, and the row sums of the unsigned array give A000311(n+1) (Schroeder's fourth problem). These sums coincide with those of the triangle A134991.
The signed a(n,k) numbers, k=1,...,p(n)=A000041(n), derive from the multinomial M_3 numbers A036040 (see also the W. Lang link there), namely, if the k-th partition of n in A-St order has exponents (enk[1],...,enk[n]) then a(n,k) = ((-1)^m)*M3(n+m, (ehatnk[1],...,ehatnk[n+m])) with m the number of parts, i.e., m:=Sum_{j=1..n} enk[j], and M3(n+m, (ehatnk[1],...,ehatnk[n+m])):=(n+m)!/(Product_{j=1..n+m} j!^ehatnk[j]*ehatnk[j]!), where the n+m exponents ehatnk are ehatnk[1]:=0, (ehatnk[2],...,ehatnk[n+1]) := (enk[1],...,enk[n]), and (ehatnk[n+1],...,ehatnk[n+m]):=(0,...,0) (i.e., m-1 zeros).
The compositional inverse of the formal power series of the e.g.f. type g(x) = Sum_{j>=1} g[j]*(x^j)/j! is f = g^[-1] with f(y) = Sum_{n>=1} f[n]*(y^n)/n!, and f[n] = fhat[n]/g[1]^(2*n-1) with fhat[1]=1 (f[1] = 1/g[1]) and f[n+1] = Sum_{k=1..p(n)} a(n,k)*g(n,k), n >= 1, where p(n) = A000041(n) (number of partitions of n), and g(n,k) is the monomial in coefficients of g(x) corresponding to the k-th partition of 2*n with n parts in A-St order. For details and a remark on the Faa di Bruno Hopf algebra see the W. Lang link.

			  -1;
  -1,  3;
  -1, 10, -15;
  -1, 15,  10, -105,  105;
  -1, 21,  35, -210, -280, 1260, -945;
...
a(4,4): 4th partition of 4 has exponents (2,1,0,0) with m=3, and the derived exponents ehatm are (0,2,1,0,0,0,0) with one leading and 2 extra trailing zeros. (4+3)!/(2!^2*2!*3!^1*1!) = 105, hence a(4,4) = ((-1)^3)*105 = -105.
fhat[4] = -1*g[1]^2*g[4] +10*g[1]*g[2]*g[3] - 15*g[2]^3 (n=3: 3 parts partitions of 6 for the g-monomials in A-St order).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 831-2.
R. Aldrovandi, Special Matrices of Mathematical Physics, World Scientific, 2001, p. 175, eq. (13.84).
Ch. A. Charalambides, Enumerative Combinatorics, Chapman &Hall/CRC, 2002, p. 437, eq. (11.43) with p. 428. eq. (11.29).

W. P. Johnson, Combinatorics of Higher Derivatives of Inverses, Amer. Math. Monthly 109 (3), (2002), 273-277
Wolfdieter Lang, E.g.f. Lagrange inversion partition array.

See the fhat[n] formula explained above, and the W. Lang link for more details.

cp's OEIS Frontend

A033282 Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.

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A134991 Triangle of Ward numbers T(n,k) read by rows.

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A112486 Coefficient triangle for polynomials used for e.g.f.s for unsigned Stirling1 diagonals.

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A112487 a(n) = Sum_{k=0..n} E2(n, k)*2^k, where E2(n, k) are the second-order Eulerian numbers A340556.

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A074060 Graded dimension of the cohomology ring of the moduli space of n-pointed stable curves of genus 0 satisfying the associativity equations of physics (also known as the WDVV equations).

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A356145 Coefficients of the inverse refined Eulerian partition polynomials [E]^{-1}, partitional inverse to A145271. Irregular triangle read by row with lengths A000041.

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