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 A019538 #443 Aug 30 2025 20:22:08
%S A019538 1,1,2,1,6,6,1,14,36,24,1,30,150,240,120,1,62,540,1560,1800,720,1,126,
%T A019538 1806,8400,16800,15120,5040,1,254,5796,40824,126000,191520,141120,
%U A019538 40320,1,510,18150,186480,834120,1905120,2328480,1451520,362880,1,1022,55980,818520,5103000,16435440,29635200,30240000,16329600,3628800
%N A019538 Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).
%C A019538 Number of ways n labeled objects can be distributed into k nonempty parcels. Also number of special terms in n variables with maximal degree k.
%C A019538 In older terminology these are called differences of 0. - _Michael Somos_, Oct 08 2003
%C A019538 Number of surjections (onto functions) from an n-element set to a k-element set.
%C A019538 Also coefficients (in ascending order) of so-called ordered Bell polynomials.
%C A019538 (k-1)!*Stirling2(n,k-1) is the number of chain topologies on an n-set having k open sets [Stephen].
%C A019538 Number of set compositions (ordered set partitions) of n items into k parts. Number of k dimensional 'faces' of the n dimensional permutohedron (see Simion, p. 162). - _Mitch Harris_, Jan 16 2007
%C A019538 Correction of comment before: Number of (n-k)-dimensional 'faces' of the permutohedron of order n (an (n-1)-dimensional polytope). - _Tilman Piesk_, Oct 29 2014
%C A019538 This array is related to the reciprocal of an e.g.f. as sketched in A133314. For example, the coefficient of the fourth-order term in the Taylor series expansion of 1/(a(0) + a(1) x + a(2) x^2/2! + a(3) x^3/3! + ...) is a(0)^(-5) * {24 a(1)^4 - 36 a(1)^2 a(2) a(0) + [8 a(1) a(3) + 6 a(2)^2] a(0)^2 - a(4) a(0)^3}. The unsigned coefficients characterize the P3 permutohedron depicted on page 10 in the Loday link with 24 vertices (0-D faces), 36 edges (1-D faces), 6 squares (2-D faces), 8 hexagons (2-D faces) and 1 3-D permutohedron. Summing coefficients over like dimensions gives A019538 and A090582. Compare to A133437 for the associahedron. - _Tom Copeland_, Sep 29 2008, Oct 07 2008
%C A019538 Further to the comments of _Tom Copeland_ above, the permutohedron of type A_3 can be taken as the truncated octahedron. Its dual is the tetrakis hexahedron, a simplicial polyhedron, with f-vector (1,14,36,24) giving the fourth row of this triangle. See the Wikipedia entry and [Fomin and Reading p. 21]. The corresponding h-vectors of permutohedra of type A give the rows of the triangle of Eulerian numbers A008292. See A145901 and A145902 for the array of f-vectors for type B and type D permutohedra respectively. - _Peter Bala_, Oct 26 2008
%C A019538 Subtriangle of triangle in A131689. - _Philippe Deléham_, Nov 03 2008
%C A019538 Since T(n,k) counts surjective functions and surjective functions are "consistent", T(n,k) satisfies a binomial identity, namely, T(n,x+y) = Sum_{j=0..n} C(n,j)*T(j,x)*T(n-j,y). For definition of consistent functions and a generalized binomial identity, see "Toy stories and combinatorial identities" in the link section below. - _Dennis P. Walsh_, Feb 24 2012
%C A019538 T(n,k) is the number of labeled forests on n+k vertices satisfying the following two conditions: (i) each forest consists of exactly k rooted trees with roots labeled 1, 2, ..., k; (ii) every root has at least one child vertex. - _Dennis P. Walsh_, Feb 24 2012
%C A019538 The triangle is the inverse binomial transform of triangle A028246, deleting the left column and shifting up one row. - _Gary W. Adamson_, Mar 05 2012
%C A019538 See A074909 for associations among this array and the Bernoulli polynomials and their umbral compositional inverses. - _Tom Copeland_, Nov 14 2014
%C A019538 E.g.f. for the shifted signed polynomials is G(x,t) = (e^t-1)/[1+(1+x)(e^t-1)] = 1-(1+x)(e^t-1) + (1+x)^2(e^t-1)^2 - ... (see also A008292 and A074909), which has the infinitesimal generator g(x,u)d/du = [(1-x*u)(1-(1+x)u)]d/du, i.e., exp[t*g(x,u)d/du]u eval. at u=0 gives G(x,t), and dG(x,t)/dt = g(x,G(x,t)). The compositional inverse is log((1-xt)/(1-(1+x)t)). G(x,t) is a generating series associated to the generalized Hirzebruch genera. See the G. Rzadowski link for the relation of the derivatives of g(x,u) to solutions of the Riccatt differential equation, soliton solns. to the KdV equation, and the Eulerian and Bernoulli numbers. In addition A145271 connects products of derivatives of g(x,u) and the refined Eulerian numbers to the inverse of G(x,t), which gives the normalized, reverse face polynomials of the simplices (A135278, divided by n+1). See A028246 for the generator g(x,u)d/dx. - _Tom Copeland_, Nov 21 2014
%C A019538 For connections to toric varieties and Eulerian polynomials, see the Dolgachev and Lunts and the Stembridge links. - _Tom Copeland_, Dec 31 2015
%C A019538 See A008279 for a relation between the e.g.f.s enumerating the faces of permutahedra (this entry) and stellahedra. - _Tom Copeland_, Nov 14 2016
%C A019538 T(n, k) appears in a Worpitzky identity relating monomials to binomials: x^n = Sum_{k=1..n} T(n, k)*binomial(x,k), n >= 1. See eq. (11.) of the Worpitzky link on p. 209. The relation to the Eulerian numbers is given there in eqs. (14.) and (15.). See the formula below relating to A008292. See also Graham et al. eq. (6.10) (relating monomials to falling factorials) on p. 248 (2nd ed. p. 262). The Worpitzky identity given in the Graham et al. reference as eq. (6.37) (2nd ed. p. 269) is eq. (5.), p. 207, of Worpitzky. - _Wolfdieter Lang_, Mar 10 2017
%C A019538 T(n, m) is also the number of minimum clique coverings and minimum matchings in the complete bipartite graph K_{m,n}. - _Eric W. Weisstein_, Apr 26 2017
%C A019538 From the Hasan and Franco and Hasan papers: The m-permutohedra for m=1,2,3,4 are the line segment, hexagon, truncated octahedron and omnitruncated 5-cell. The first three are well-known from the study of elliptic models, brane tilings and brane brick models. The m+1 torus can be tiled by a single (m+2)-permutohedron. Relations to toric Calabi-Yau Kahler manifolds are also discussed. - _Tom Copeland_, May 14 2020
%C A019538 From _Manfred Boergens_, Jul 25 2021: (Start)
%C A019538 Number of n X k binary matrices with row sums = 1 and no zero columns. These matrices are a subset of the matrices defining A183109.
%C A019538 The distribution into parcels in the leading comment can be regarded as a covering of [n] by tuples (A_1,...,A_k) in P([n])^k with nonempty and disjoint A_j, with P(.) denoting the power set (corrected for clarity by _Manfred Boergens_, May 26 2024). For the non-disjoint case see A183109 and A218695.
%C A019538 For tuples with "nonempty" dropped see A089072. For tuples with "nonempty and disjoint" dropped see A092477 and A329943 (amendment by _Manfred Boergens_, Jun 24 2024). (End)
%D A019538 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 89, ex. 1; also p. 210.
%D A019538 Miklos Bona, Combinatorics of Permutations, Chapman and Hall,2004, p.12.
%D A019538 G. Boole, A Treatise On The Calculus of Finite Differences, Dover Publications, 1960, p. 20.
%D A019538 H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 212.
%D A019538 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, 1989, p. 155. Also eqs.(6.10) and (6.37).
%D A019538 Kiran S. Kedlaya and Andrew V. Sutherland, Computing L -Series of Hyperelliptic Curves in Algorithmic Number Theory Lecture Notes in Computer Science Volume 5011/2008.
%D A019538 T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Section 5.6.
%D A019538 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 33.
%D A019538 J. F. Steffensen, Interpolation, 2nd ed., Chelsea, NY, 1950, see p. 54.
%D A019538 A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911, p. 31.
%D A019538 E. Whittaker and G. Robinson, The Calculus of Observations, Blackie, London, 4th ed., 1949; p. 7.
%H A019538 T. D. Noe, <a href="/A019538/b019538.txt">Rows n = 1..100 of triangle, flattened</a>
%H A019538 N. Arkani-Hamed, Y. Bai, S. He, and G. Yan, <a href="https://arxiv.org/abs/1711.09102">Scattering forms and the positive geometry of kinematics, color, and the worldsheet </a>, arXiv:1711.09102 [hep-th], 2017. [From _Tom Copeland_, Jun 24 2018]
%H A019538 Mohammad K. Azarian, <a href="https://doi.org/10.12988/imf.2022.912321">Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions</a>, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141. (See Theorem 2.1 (iv), p. 131.)
%H A019538 J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, <a href="http://arxiv.org/abs/1307.2010">Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure</a>, arXiv:1307.2010 [math.CO], 2013.
%H A019538 J. F. Barbero G., J. Salas and E. J. S. Villaseñor, <a href="http://arxiv.org/abs/1307.5624">Bivariate Generating Functions for a Class of Linear Recurrences. II. Applications</a>, arXiv preprint arXiv:1307.5624 [math.CO], 2013.
%H A019538 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 A019538 Moussa Benoumhani, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Benoumhani/benoumhani11.html">The Number of Topologies on a Finite Set</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.
%H A019538 Olivier Bodini, Antoine Genitrini, Cécile Mailler, and Mehdi Naima, <a href="https://hal.archives-ouvertes.fr/hal-02865198">Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study</a>, hal-02865198 [math.CO] / [math.PR] / [cs.DS] / [cs.DM], 2020.
%H A019538 K. N. Boyadzhiev, <a href="http://www.emis.de/journals/HOA/IJMMS/Volume2005_23/792107.abs.html">A Series transformation formula and related polynomials</a>, Int. J. Math. Math. Sc. 23 (2005), 3849-3866.
%H A019538 Xavier Bultel, Jannik Dreier, Matthieu Giraud, Marie Izaute, Timothée Kheyrkhah, Pascal Lafourcade, Dounia Lakhzoum, Vincent Marlin, and Ladislav Motá, <a href="https://hal.archives-ouvertes.fr/hal-01777898">Security Analysis and Psychological Study of Authentication Methods with PIN Codes</a>, RCIS 2018 - IEEE 12th International Conference on Research Challenges in Information Science, 2018.
%H A019538 J. L. Chandon, J. LeMaire, and J. Pouget, <a href="http://www.numdam.org/item?id=MSH_1978__62__61_0">Dénombrement des quasi-ordres sur un ensemble fini</a>, Math. Sci. Humaines, No. 62 (1978), 61-80.
%H A019538 Frédéric Chapoton and Vincent Pilaud, <a href="https://arxiv.org/abs/2201.06896">Shuffles of deformed permutahedra, multiplihedra, constrainahedra, and biassociahedra</a>, arXiv:2201.06896 [math.CO], 2022. See p. 12, 19.
%H A019538 Tom Copeland, <a href="http://tcjpn.wordpress.com/2014/12/14/hirzebruch-and-the-generalized-todd-class/">The Bernoulli polynomials and Hirzebruch's generalized Todd class</a>
%H A019538 E. Delucchi, A. Pixton, and L. Sabalka, <a href="http://dx.doi.org/10.1016/j.disc.2011.08.032">Face vectors of subdivided simplicial complexes</a> Discrete Math. 312 (2012), no. 2, 248--257. MR2852583 (2012j:05470). See p. 250. - _N. J. A. Sloane_, Apr 04 2014
%H A019538 I. Dolgachev and V. Lunts, <a href="http://www.math.lsa.umich.edu/~idolga/luntz94.pdf">A character formula for the representation of the Weyl group in the cohomology of the associated toric variety</a>, Journal of Algebra, 168, 741-772, (1994).
%H A019538 Loïc Foissy and Frédéric Patras, <a href="https://hal.science/hal-04773293">Lie theory of free cocommutative and commutative cofree Hopf algebras</a>, hal-04773293, 2024. See p. 42.
%H A019538 S. Fomin and N. 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.
%H A019538 S. Franco and A. Hasan, <a href="https://arxiv.org/abs/1904.07954">Graded Quivers, Generalized Dimer Models and Toric Geometry </a>, arXiv preprint arXiv:1904.07954 [hep-th], 2019.
%H A019538 T. L. Friedman and P. Klingsberg, <a href="http://dx.doi.org/10.1155/IJMMS/2006/96327">Combinatorial identities by way of Wilf's multigraph model</a>, Int. J. Math. Math. Sci. (2006) 96327, defn. 4.3.
%H A019538 X. Gao, S. He, and Y. Zhan, <a href="https://arxiv.org/abs/1708.08701">Labelled tree graphs, Feynman diagrams and disk integrals </a>, arXiv:1708.08701 [hep-th], 2017. [From _Tom Copeland_, Jun 24 2018]
%H A019538 M. Goebel, <a href="http://www.informatik.uni-trier.de/~ley/db/journals/aaecc/aaecc8.html">On the number of special permutation-invariant orbits and terms</a>, in Applicable Algebra in Engin., Comm. and Comp. (AAECC 8), Volume 8, Number 6, 1997, pp. 505-509 (Lect. Notes Comp. Sci.)
%H A019538 M. Griffiths and I. Mezo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Griffiths/griffiths11.html">A generalization of Stirling Numbers of the Second Kind via a special multiset</a>, JIS 13 (2010) #10.2.5
%H A019538 J. Gubeladze and J. Love, <a href="http://arxiv.org/abs/1304.3775">Vertex maps between simplices, cubes, and crosspolytopes</a>, arXiv preprint arXiv:1304.3775 [math.CO], 2013.
%H A019538 Jens Gulin and Kalle Åström, <a href="https://lucris.lub.lu.se/ws/portalfiles/portal/196591300/IPINwip_ARMPIT_final.pdf">Alternative implementations of the Auxiliary Duplicating Permutation Invariant Training</a>, Proc Work-in-Progress Papers at 14th Int'l Conf. Indoor Positioning Indoor Nav. (IPIN-WiP 2024). See p. 6.
%H A019538 Li Guo, <a href="http://arxiv.org/abs/math.AC/0402348">Baxter algebras, Stirling numbers and partitions</a>, arXiv:math/0402348 [math.AC], 2004.
%H A019538 M. Iida, <a href="http://libir.josai.ac.jp/infolib/user_contents/pdf/JOS-13447777-05_61.pdf">On Triangle of numbers</a>, Josai Mathematical Monographs, Vol. 5 (2012), 61-70;
%H A019538 A. Hasan, <a href="https://academicworks.cuny.edu/gc_etds/3321/">Physics and Mathematics of Graded Quivers</a>, dissertation, Graduate Center, City University of New York, 2019.
%H A019538 Gábor Hetyei, <a href="http://dx.doi.org/10.1007/s00454-005-1190-2">The Stirling polynomial of a simplicial complex</a> Discrete and Computational Geometry 35, Number 3, March 2006, pp 437-455.
%H A019538 Gábor Hetyei, <a href="https://math.charlotte.edu/preprint-archive/face-enumeration-using-generalized-binomial-coefficients">Face enumeration using generalized binomial coefficients</a>. Online draft version of the Hetyei paper referenced above.
%H A019538 Sergey Kitaev, Andrew Niedermaier, Jeffrey Remmel, and Manda Riehl, <a href="https://doi.org/10.1155/2013/634823">Generalized pattern-matching conditions for Ck wreath Sn</a>, ISRN Comb. 2013, Article ID 634823, 20 p. (2013), eq. (87)
%H A019538 Germain Kreweras, <a href="http://www.numdam.org/item?id=MSH_1963__3__31_0">Une dualité élémentaire souvent utile dans les problèmes combinatoires</a>, Mathématiques et Sciences Humaines 3 (1963): 31-41.
%H A019538 Germain Kreweras, <a href="http://www.numdam.org/item?id=MSH_1976__53__5_0">Les préordres totaux compatibles avec un ordre partiel</a>, Math. Sci. Humaines No. 53 (1976), 5-30.
%H A019538 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 A019538 L. Liu and Y. Wang, <a href="http://www.arXiv.org/abs/math.CO/0509207">A unified approach to polynomial sequences with only real zeros</a>, arXiv:math/0509207 [math.CO], 2005-2006.
%H A019538 J. Loday, <a href="https://web.archive.org/web/20100202074906/http://www-irma.u-strasbg.fr/~loday/PAPERS/MultFAsENG2.pdf">The Multiple Facets of the Associahedron</a> [From _Tom Copeland_, Sep 29 2008]
%H A019538 František Marko, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Marko/marko5.html">Sums of Consecutive Powers as a Linear Combination of Products of Two Figurate Numbers</a>, J. Int. Seq., Vol. 25 (2022), Article 22.2.8.
%H A019538 Richard J. Mathar, <a href="https://arxiv.org/abs/2306.08022">On an Alternating Double Sum of a Triple Product of Aerated Binomial Coefficients</a>, arXiv:2306.08022 [math.GM], 2023.
%H A019538 MathOverflow, <a href="http://mathoverflow.net/questions/29490/how-many-surjections-are-there-from-a-set-of-size-n">How many surjections are there from a set of size n?</a> [From _Tom Copeland_, Sep 09 2015]
%H A019538 E. Mendelson, <a href="http://www.jstor.org/stable/2690085">Races with Ties</a>, Math. Mag. 55 (1982), 170-175.
%H A019538 Lucas Chaves Meyles, Pamela E. Harris, Richter Jordaan, Gordon Rojas Kirby, Sam Sehayek, and Ethan Spingarn, <a href="https://arxiv.org/abs/2305.15554">Unit-Interval Parking Functions and the Permutohedron</a>, arXiv:2305.15554 [math.CO], 2023.
%H A019538 T. S. Motzkin, <a href="/A000262/a000262.pdf">Sorting numbers for cylinders and other classification numbers</a>, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
%H A019538 Arthur Nunge, <a href="https://arxiv.org/abs/1805.01797">Eulerian polynomials on segmented permutations</a>, arXiv:1805.01797 [math.CO], 2018.
%H A019538 OEIS Wiki, <a href="http://oeis.org/wiki/Sorting_numbers">Sorting numbers</a>
%H A019538 K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0312202">Hierarchical Dobinski-type relations via substitution and the moment problem</a>, arXiv:quant-ph/0312202, 2003.
%H A019538 Vincent Pilaud and V. Pons, <a href="http://arxiv.org/abs/1606.09643">Permutrees</a>, arXiv preprint arXiv:1606.09643 [math.CO], 2016-2017.
%H A019538 P. A. Piza, <a href="http://www.jstor.org/stable/3029339">Kummer numbers</a>, Mathematics Magazine, 21 (1947/1948), 257-260.
%H A019538 P. A. Piza, <a href="/A001117/a001117.pdf">Kummer numbers</a>, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy]
%H A019538 G. Rzadkowski, <a href="http://dx.doi.org/10.1142/S1402925110000635">Bernoulli numbers and solitons revisited</a>, Journal of Nonlinear Mathematical Physics, 17:1, 121-126, DOI: 10.1142/S1402925110000635
%H A019538 Natalie Schluter, <a href="https://openreview.net/pdf?id=rJTboukPG">The Taylor expansion for dropout is divergent</a>, 2018.
%H A019538 Natalie Schluter, <a href="https://arxiv.org/abs/1905.11320">On approximating dropout noise injection</a>, arXiv:1905.11320 [cs.LG], 2019.
%H A019538 Mark Shattuck, <a href="https://www.researchgate.net/publication/269168303_Combinatorial_proofs_of_some_Stirling_number_formulas">Combinatorial proofs of some Stirling number formulas</a>, Preprint (ResearchGate), 2014.
%H A019538 John K. Sikora, <a href="https://arxiv.org/abs/1806.00887">On Calculating the Coefficients of a Polynomial Generated Sequence Using the Worpitzky Number Triangles</a>, arXiv:1806.00887 [math.NT], 2018.
%H A019538 R. 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 A019538 J. Stembridge, <a href="http://deepblue.lib.umich.edu/bitstream/handle/2027.42/31472/0000394.pdf?sequence=1">Some permutation representations of Weyl groups associated with the cohomology of toric varieties</a>, Adv. in Math., 106, 244-301, (1994).
%H A019538 D. Stephen, <a href="http://www.jstor.org/stable/2315186">Topology on Finite Sets</a>, American Mathematical Monthly, 75: 739 - 741, 1968.
%H A019538 S. M. Tanny, <a href="https://doi.org/10.4153/CMB-1974-132-8">On some numbers related to the Bell numbers</a>, Canad. Math. Bull. Vol. 17 (5), 733-738, 1975.
%H A019538 A. H. Voigt, <a href="/A000918/a000918.pdf">Theorie der Zahlenreihen und der Reihengleichungen</a>, Goschen, Leipzig, 1911. [Annotated scans of pages 30-33 only]
%H A019538 D. P. Walsh, <a href="http://frank.mtsu.edu/~dwalsh/ATHEORM4.pdf">Toy stories and combinatorial identities</a>
%H A019538 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CliqueCovering.html">Clique Covering</a>
%H A019538 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>
%H A019538 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Matching.html">Matching</a>
%H A019538 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MinimumEdgeCover.html">Minimum Edge Cover</a>
%H A019538 Wikipedia, <a href="http://en.wikipedia.org/wiki/Truncated_octahedron">Truncated octahedron</a>
%H A019538 J. Worpitzky, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002158698">Studien über die Bernoullischen und Eulerschen Zahlen</a>, J. reine u. angew. Math., 94 (1883) 203-232.
%F A019538 T(n, k) = k*(T(n-1, k-1)+T(n-1, k)) with T(0, 0) = 1 [or T(1, 1) = 1]. - _Henry Bottomley_, Mar 02 2001
%F A019538 E.g.f.: (y*(exp(x)-1) - exp(x))/(y*(exp(x)-1) - 1). - _Vladeta Jovovic_, Jan 30 2003
%F A019538 Equals [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ...] where DELTA is Deléham's operator defined in A084938.
%F A019538 T(n, k) = Sum_{j=0..k} (-1)^(k-j)*j^n*binomial(k, j). - Mario Catalani (mario.catalani(AT)unito.it), Nov 28 2003. See Graham et al., eq. (6.19), p. 251. For a proof see _Bert Seghers_, Jun 29 2013.
%F A019538 Sum_{k=0..n} T(n, k)(-1)^(n-k) = 1, Sum_{k=0..n} T(n, k)(-1)^k = (-1)^n. - Mario Catalani (mario.catalani(AT)unito.it), Dec 11 2003
%F A019538 O.g.f. for n-th row: polylog(-n, x/(1+x))/(x+x^2). - _Vladeta Jovovic_, Jan 30 2005
%F A019538 E.g.f.: 1 / (1 + t*(1-exp(x))). - _Tom Copeland_, Oct 13 2008
%F A019538 From _Peter Bala_, Oct 26 2008: (Start)
%F A019538 O.g.f. as a continued fraction: 1/(1 - x*t/(1 - (x + 1)*t/(1 - 2*x*t/(1 - 2*(x + 1)*t/(1 - ...))))) = 1 + x*t + (x + 2*x^2)*t^2 + (x + 6*x^2 + 6*x^3)*t^3 + ... .
%F A019538 The row polynomials R(n,x), which begin R(1,x) = x, R(2,x) = x + 2*x^2, R(3,x) = x + 6*x^2 + 6*x^3, satisfy the recurrence x*d/dx ((x + 1)*R(n,x)) = R(n+1,x). It follows that the zeros of R(n,x) are real and negative (apply Corollary 1.2 of [Liu and Wang]).
%F A019538 Since this is the triangle of f-vectors of the (simplicial complexes dual to the) type A permutohedra, whose h-vectors form the Eulerian number triangle A008292, the coefficients of the polynomial (x-1)^n*R(n,1/(x-1)) give the n-th row of A008292. For example, from row 3 we have x^2 + 6*x + 6 = 1 + 4*y + y^2, where y = x + 1, producing [1,4,1] as the third row of A008292. The matrix product A008292 * A007318 gives the mirror image of this triangle (see A090582).
%F A019538 For n,k >= 0, T(n+1,k+1) = Sum_{j=0..k} (-1)^(k-j)*binomial(k,j)*[(j+1)^(n+1) - j^(n+1)]. The matrix product of Pascal's triangle A007318 with the current array gives (essentially) A047969. This triangle is also related to triangle A047969 by means of the S-transform of [Hetyei], a linear transformation of polynomials whose value on the basis monomials x^k is given by S(x^k) = binomial(x,k). The S-transform of the shifted n-th row polynomial Q(n,x) := R(n,x)/x is S(Q(n,x)) = (x+1)^n - x^n. For example, from row 3 we obtain S(1 + 6*x + 6*x^2) = 1 + 6*x + 6*x*(x-1)/2 = 1 + 3*x + 3*x^2 = (x+1)^3 - x^3. For fixed k, the values S(Q(n,k)) give the nonzero entries in column (k-1) of the triangle A047969 (the Hilbert transform of the Eulerian numbers). (End)
%F A019538 E.g.f.: (exp(x)-1)^k = sum T(n,k)x^n/n!. - _Vladimir Kruchinin_, Aug 10 2010
%F A019538 T(n,k) = Sum_{i=1..k} A(n,i)*Binomial(n-i,k-i) where A(n,i) is the number of n-permutations that have i ascending runs, A008292.
%F A019538 From _Tom Copeland_, Oct 11 2011: (Start)
%F A019538 With e.g.f. A(x,t) = -1 + 1/(1+t*(1-exp(x))), the comp. inverse in x is B(x,t) = log(((1+t)/t) - 1/(t(1+x))).
%F A019538 With h(x,t) = 1/(dB/dx)= (1+x)((1+t)(1+x)-1), the row polynomial P(n,t) is given by (h(x,t)*d/dx)^n x, eval. at x=0, A=exp(x*h(y,t)*d/dy) y, eval. at y=0, and dA/dx = h(A(x,t),t), with P(0,t)=0.
%F A019538 (A factor of -1/n! was removed by Copeland on Aug 25 2016.) (End)
%F A019538 The term linear in x of [x*h(d/dx,t)]^n 1 gives the n-th row polynomial. (See A134685.) - _Tom Copeland_, Nov 07 2011
%F A019538 Row polynomials are given by D^n(1/(1-x*t)) evaluated at x = 0, where D is the operator (1+x)*d/dx. - _Peter Bala_, Nov 25 2011
%F A019538 T(n,x+y) = Sum_{j=0..n} binomial(n,j)*T(j,x)*T(n-j,y). - _Dennis P. Walsh_, Feb 24 2012
%F A019538 Let P be a Rota-Baxter operator of weight 1 satisfying the identity P(x)*P(y) = P(P(x)*y) + P(x*P(y)) + P(x*y). Then P(1)^2 = P(1) + 2*P^2(1). More generally, Guo shows that P(1)^n = Sum_{k=1..n} T(n,k)*P^k(1). - _Peter Bala_, Jun 08 2012
%F A019538 Sum_{i=1..n} (-1)^i*T(n,i)/i = 0, for n > 1. - _Leonid Bedratyuk_, Aug 09 2012
%F A019538 T(n, k) = Sum_{j=0..k} (-1)^j*binomial(k, j)*(k-j)^n. [M. Catalani's re-indexed formula from Nov 28 2003] Proof: count the surjections of [n] onto [k] with the inclusion-exclusion principle, as an alternating sum of the number of functions from [n] to [k-j]. - _Bert Seghers_, Jun 29 2013
%F A019538 n-th row polynomial = 1/(1 + x)*( Sum_{k>=0} k^n*(x/(1 + x))^k ), valid for x in the open interval (-1/2, inf). See Tanny link. Cf. A145901. - _Peter Bala_, Jul 22 2014
%F A019538 T(n,k) = k * A141618(n,k-1) / binomial(n,k-1). - _Tom Copeland_, Oct 25 2014
%F A019538 Sum_{n>=0} n^k*a^n = Sum_{i=1..k} (a / (1 - a))^i * T(k, i)/(1-a) for |a| < 1. - _David A. Corneth_, Mar 09 2015
%F A019538 From _Peter Bala_, May 26 2015: (Start)
%F A019538 The row polynomials R(n,x) satisfy (1 + x)*R(n,x) = (-1)^n*x*R(n,-(1 + x)).
%F A019538 For a fixed integer k, the expansion of the function A(k,z) := exp( Sum_{n >= 1} R(n,k)*z^n/n ) has integer coefficients and satisfies the functional equation A(k,z)^(k + 1) = BINOMIAL(A(k,z))^k, where BINOMIAL(F(z))= 1/(1 - z)*F(z/(1 - z)) denotes the binomial transform of the o.g.f. F(z). Cf. A145901. For cases see A084784 (k = 1), A090352 (k = 2), A090355 (k = 3), A090357 (k = 4), A090362 (k = 5) and A084785 (k = -2 with z -> -z).
%F A019538 A(k,z)^(k + 1) = A(-(k + 1),-z)^k and hence BINOMIAL(A(k,z)) = A(-(k + 1),-z). (End)
%F A019538 From _Tom Copeland_, Oct 19 2016: (Start)
%F A019538 Let a(1) = 1 + x + B(1) = x + 1/2 and a(n) = B(n) = (B.)^n, where B(n) are the Bernoulli numbers defined by e^(B.t) = t / (e^t-1), then t / e^(a.t) = t / [(x + 1) * t + exp(B.t)] = (e^t - 1) /[ 1 + (x + 1) (e^t - 1)] = exp(p.(x)t), where (p.(x))^n = p_n(x) are the shifted, signed row polynomials of this array: p_0(x) = 0, p_1(x) = 1, p_2(x) = -(1 + 2 x), p_3(x) = 1 + 6 x + 6 x^2, ... and p_n(x) = n * b(n-1), where b(n) are the partition polynomials of A133314 evaluated with these a(n).
%F A019538 Sum_{n > 0} R(n,-1/2) x^n/n! = 2 * tanh(x/2), where R(n,x) = Sum_{k = 1..n} T(n,k) x^(k-1) are the shifted row polynomials of this entry, so R(n,-1/2) = 4 * (2^(n+1)-1) B(n+1)/(n+1). (Cf. A000182.)
%F A019538 (End)
%F A019538 Also the Bernoulli numbers are given by B(n) = Sum_{k =1..n} (-1)^k T(n,k) / (k+1). - _Tom Copeland_, Nov 06 2016
%F A019538 G.f. for column k: k! x^k / Product_{i=1..k} (1-i*x). - _Robert A. Russell_, Sep 25 2018
%F A019538 a(j) <= A183109(j). - _Manfred Boergens_, Jul 25 2021
%e A019538 The triangle T(n, k) begins:
%e A019538 n\k 1 2 3 4 5 6 7 8 9 10
%e A019538 1: 1
%e A019538 2: 1 2
%e A019538 3: 1 6 6
%e A019538 4: 1 14 36 24
%e A019538 5: 1 30 150 240 120
%e A019538 6: 1 62 540 1560 1800 720
%e A019538 7: 1 126 1806 8400 16800 15120 5040
%e A019538 8: 1 254 5796 40824 126000 191520 141120 40320
%e A019538 9: 1 510 18150 186480 834120 1905120 2328480 1451520 362880
%e A019538 10: 1 1022 55980 818520 5103000 16435440 29635200 30240000 16329600 3628800
%e A019538 ... Reformatted and extended - _Wolfdieter Lang_, Oct 04 2014
%e A019538 ---------------------------------------------------------------------------
%e A019538 T(4,1) = 1: {1234}. T(4,2) = 14: {1}{234} (4 ways), {12}{34} (6 ways), {123}{4} (4 ways). T(4,3) = 36: {12}{3}{4} (12 ways), {1}{23}{4} (12 ways), {1}{2}{34} (12 ways). T(4,4) = 1: {1}{2}{3}{4} (1 way).
%p A019538 with(combinat): A019538 := (n,k)->k!*stirling2(n,k);
%t A019538 Table[k! StirlingS2[n, k], {n, 9}, {k, n}] // Flatten
%o A019538 (PARI) {T(n, k) = if( k<0 || k>n, 0, sum(i=0, k, (-1)^i * binomial(k, i) * (k-i)^n))}; /* _Michael Somos_, Oct 08 2003 */
%o A019538 (Haskell)
%o A019538 a019538 n k = a019538_tabl !! (n-1) !! (k-1)
%o A019538 a019538_row n = a019538_tabl !! (n-1)
%o A019538 a019538_tabl = iterate f [1] where
%o A019538 f xs = zipWith (*) [1..] $ zipWith (+) ([0] ++ xs) (xs ++ [0])
%o A019538 -- _Reinhard Zumkeller_, Dec 15 2013
%o A019538 (Sage) def T(n, k): return factorial(k)*stirling_number2(n,k) # _Danny Rorabaugh_, Oct 10 2015
%Y A019538 Row sums give A000670. Maximal terms in rows give A002869. Central terms T(2k-1,k) give A233734.
%Y A019538 Diagonal is n! (A000142). 2nd diagonal is A001286. 3rd diagonal is A037960.
%Y A019538 Cf. A000918, A000919, A001117, A001118, A008275, A008277, A008279, A048594, A059117, A059515, A074909, A084938, A089072, A092477, A183109, A218695, A329943.
%Y A019538 Reflected version of A090582. A371568 is another version.
%Y A019538 See also the two closely related triangles: A008277(n, k) = T(n, k)/k! (Stirling numbers of second kind) and A028246(n, k) = T(n, k)/k.
%Y A019538 Cf. A033282 'faces' of the associahedron.
%Y A019538 Cf. A008292, A047969, A145901, A145902. - _Peter Bala_, Oct 26 2008
%Y A019538 Visible in the 3-D array in A249042.
%Y A019538 See also A000182.
%K A019538 nonn,tabl,easy,nice,changed
%O A019538 1,3
%A A019538 _N. J. A. Sloane_ and Manfred Goebel (goebel(AT)informatik.uni-tuebingen.de), Dec 11 1996