login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A008292 Triangle of Eulerian numbers T(n,k) (n >= 1, 1 <= k <= n) read by rows. 277

%I

%S 1,1,1,1,4,1,1,11,11,1,1,26,66,26,1,1,57,302,302,57,1,1,120,1191,2416,

%T 1191,120,1,1,247,4293,15619,15619,4293,247,1,1,502,14608,88234,

%U 156190,88234,14608,502,1,1,1013,47840,455192,1310354,1310354,455192,47840,1013

%N Triangle of Eulerian numbers T(n,k) (n >= 1, 1 <= k <= n) read by rows.

%C The indexing used here follows that given in the classic books by Riordan and Comtet. For two other versions see A173018 and A123125. - _N. J. A. Sloane_, Nov 21 2010

%C Coefficients of Eulerian polynomials. Number of permutations of n objects with k-1 rises. Number of increasing rooted trees with n+1 nodes and k leaves.

%C T(n,k)=number of permutations of [n] with k runs. T(n,k)=number of permutations of [n] requiring k readings (see the Knuth reference). T(n,k)=number of permutations of [n] having k distinct entries in its inversion table. - _Emeric Deutsch_, Jun 09 2004

%C T(n,k) = number of ways to write the Coxeter element s_{e1}s_{e1-e2}s_{e2-e3}s_{e3-e4}...s_{e_{n-1}-e_n} of the reflection group of type B_n, using s_{e_k} and as few reflections of the form s_{e_i+e_j}, where i = 1, 2, ..., n and j is not equal to i, as possible. - Pramook Khungurn (pramook(AT)mit.edu), Jul 07 2004

%C Subtriangle for k>=1 and n>=1 of triangle A123125. - _Philippe Deléham_, Oct 22 2006

%C T(n,k)/n! also represents the n-dimensional volume of the portion of the n-dimensional hypercube cut by the (n-1)-dimensional hyperplanes x_1 + x_2 + ... x_n = k, x_1 + x_2 + ... x_n = k-1; or, equivalently, it represents the probability that the sum of n independent random variables with uniform distribution between 0 and 1 is between k-1 and k. - Stefano Zunino, Oct 25 2006

%C [E(.,t)/(1-t)]^n = n!*Lag[n,-P(.,t)/(1-t)] and [-P(.,t)/(1-t)]^n = n!*Lag[n, E(.,t)/(1-t)] umbrally comprise a combinatorial Laguerre transform pair, where E(n,t) are the Eulerian polynomials and P(n,t) are the polynomials in A131758. - _Tom Copeland_, Sep 30 2007

%C From _Tom Copeland_, Oct 07 2008: (Start)

%C G(x,t) = 1/(1 + (1-exp(x*t))/t) = 1 + 1*x + (2+t)*x^2/2! + (6+6*t+t^2)*x^3/3! + ...

%C gives row polynomials for A090582, the reverse f-polynomials for the permutohedra (see A019538).

%C G(x,t-1) = 1 + 1*x + (1+t)*x^2/2! + (1+4*t+t^2)*x^3/3! + ...

%C gives row polynomials for A008292, the h-polynomials for permutohedra.

%C G((t+1)*x,-1/(t+1)) = 1 + (1+t)*x + (1+3*t+2*t^2)*x^2/2! + ...

%C gives row polynomials for A028246.

%C (End)

%C A subexceedant function f on [n] is a map f:[n] -> [n] such that 1 <= f(i) <= i for all i, 1 <= i <= n. T(n,k) equals the number of subexceedant functions f of [n] such that the image of f has cardinality k [Mantaci & Rakotondrajao]. Example T(3,2) = 4: if we identify a subexceedant function f with the word f(1)f(2)...f(n) then the subexceedant functions on [3] are 111, 112, 113, 121, 122 and 123 and four of these functions have an image set of cardinality 2. - _Peter Bala_, Oct 21 2008

%C Further to the comments of _Tom Copeland_ above, the n-th row of this triangle is the h-vector of the simplicial complex dual to a permutohedron of type A_(n-1). The corresponding f-vectors are the rows of A019538. For example, 1 + 4*x + x^2 = y^2 + 6*y + 6 and 1 + 11*x + 11*x^2 + x^3 = y^3 + 14*y^2 + 36*y + 24, where x = y + 1, give [1,6,6] and [1,14,36,24] as the third and fourth rows of A019538. The Hilbert transform of this triangle (see A145905 for the definition) is A047969. See A060187 for the triangle of Eulerian numbers of type B (the h-vectors of the simplicial complexes dual to permutohedra of type B). See A066094 for the array of h-vectors of type D. For tables of restricted Eulerian numbers see A144696 - A144699. - _Peter Bala_, Oct 26 2008

%C For a natural refinement of A008292 with connections to compositional inversion and iterated derivatives, see A145271. - _Tom Copeland_, Nov 06 2008

%C The polynomials E(z,n) = numerator(Sum_{k>=1} (-1)^(n+1)*k^n*z^(k-1)) for n >=1 lead directly to the triangle of Eulerian numbers. - _Johannes W. Meijer_, May 24 2009

%C From Walther Janous (walther.janous(AT)tirol.com), Nov 01 2009: (Start)

%C The (Eulerian) polynomials e(n,x) = Sum_{k=0..n-1} T(n,k+1)*x^k turn out to be also the numerators of the closed-form expressions of the infinite sums:

%C S(p,x):= Sum_{j>=0}(j+1)^p*x^j, that is

%C S(p,x) = e(p,x)/(1-x)^(p+1), whenever |x| < 1 and p is a positive integer.

%C (Note the inconsistent use of T(n,k) in the section listing the formula section. I adhere tacitly to the first one.) (End)

%C If n is an odd prime, then all numbers of the (n-2)-th and (n-1)-th rows are in the progression k*n+1. - _Vladimir Shevelev_, Jul 01 2011

%C The Eulerian triangle is an element of the formula for the r-th successive summation of Sum_{k=1..n} k^j which appears to be Sum_{k=1..n} T(j,k-1) * binomial(j-k+n+r, j+r). - _Gary Detlefs_, Nov 11 2011

%C Li and Wong show that T(n,k) counts the combinatorially inequivalent star polygons with n+1 vertices and sum of angles (2*k-n-1)*Pi. An equivalent formulation is: define the total sign change S(p) of a permutation p in the symmetric group S_n to be equal to Sum_{i=1..n} sign(p(i)-p(i+1)), where we take p(n+1) = p(1). T(n,k) gives the number of permutations q in S_(n+1) with q(1) = 1 and S(q) = 2*k-n-1. For example, T(3,2) = 4 since in S_4 the permutations (1243), (1324), (1342) and (1423) have total sign change 0. - _Peter Bala_, Dec 27 2011

%C Xiong, Hall and Tsao refer to Riordan and mention that a traditional Eulerian number A(n,k) is the number of permutations of (1,2...n) with k weak exceedances. - _Susanne Wienand_, Aug 25 2014

%C Connections to algebraic geometry/topology and characteristic classes are discussed in the Buchstaber and Bunkova, the Copeland, the Hirzebruch, the Lenart and Zainoulline, the Losev and Manin, and the Sheppeard links; to the Grassmannian, in the Copeland, the Farber and Postnikov, the Sheppeard, and the Williams links; and to compositional inversion and differential operators, in the Copeland and the Parker links. - _Tom Copeland_, Oct 20 2015

%C The bivariate e.g.f. noted in the formulas is related to multiplying edges in certain graphs discussed in the Paoli-Marcolli link. See p. 42. - _Tom Copeland_, Dec 18 2016

%C Distribution of left children in treeshelves is given by a shift of the Eulerian numbers. Treeshelves are ordered binary (0-1-2) increasing trees where every child is connected to its parent by a left or a right link. See A278677, A278678 or A278679 for more definitions and examples. - _Sergey Kirgizov_, Dec 24 2016

%C The row polynomial P(n, x) = Sum_{k=1..n} T(n, k)*x^k appears in the numerator of the o.g.f. G(n, x) = Sum_{m>=0} S(n, m)*x^m with S(n, m) = Sum_{j=0..m} j^n for n >= 1 as G(n, x) = Sum_{k=1..n} P(n, x)/(1 - x)^(n+2) for n >= 0 (with 0^0=1). See also triangle A131689 with a Mar 31 2017 comment for a rewritten form. For the e.g.f see A028246 with a Mar 13 2017 comment. - Wolfdieter Lang, Mar 31 2017.

%C For relations to Ehrhart polynomials, volumes of polytopes, polylogarithms, the Todd operator, and other special functions, polynomials, and sequences, see A131758 and the references therein. - _Tom Copeland_, Jun 20 2017

%C For relations to values of the Riemann zeta function at integral arguments, see A131758 and the Dupont reference. - _Tom Copeland_, Mar 19 2018

%D Mohammad K. Azarian, Geometric Series, Problem 329, Mathematics and Computer Education, Vol. 30, No. 1, Winter 1996, p. 101. Solution published in Vol. 31, No. 2, Spring 1997, pp. 196-197.

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 243.

%D F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 260.

%D D. Foata, Distributions eulériennes et mahoniennes sur le groupe des permutations, pp. 27-49 of M. Aigner, editor, Higher Combinatorics, Reidel, Dordrecht, Holland, 1977.

%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 254; 2nd. ed., p. 268.

%D D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1998, Vol. 3, p. 47 (exercise 5.1.4 Nr. 20) and p. 605 (solution).

%D Anthony Mendes and Jeffrey Remmel, Generating functions from symmetric functions, Preliminary version of book, available from Jeffrey Remmel's home page http://math.ucsd.edu/~remmel/

%D T. K. Petersen, Eulerian Numbers, Birkhauser, 2015.

%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 215.

%D R. Sedgewick and P. Flajolet, An Introduction to the Analysis of Algorithms, Addison-Wesley, Reading, MA, 1996.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Figure M3416, Academic Press, 1995.

%D H. S. Wall, Analytic Theory of Continued Fractions, Chelsea, 1973, see p. 208.

%H T. D. Noe, <a href="/A008292/b008292.txt">Rows 1 to 100 of triangle, flattened.</a>

%H V. E. Adler, <a href="http://arxiv.org/abs/1510.02900">Set partitions and integrable hierarchies</a>, arXiv:1510.02900 [nlin.SI], 2015.

%H P. Aluffi and M. Marcolli, <a href="http://arxiv.org/abs/0907.3225">Feynman motives and deletion-contraction</a>, arXiv:0907.3225 [math-ph], 2009.

%H E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce <a href="http://arxiv.org/abs/1508.03673">A generalization of Eulerian numbers via rook placements</a>, arXiv:1508.03673 [math.CO], 2015.

%H J. Fernando Barbero G., Jesús Salas, 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 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 Jean-Luc Baril, Sergey Kirgizov, Vincent Vajnovszki, <a href="https://arxiv.org/abs/1611.07793">Patterns in treeshelves</a>, arXiv:1611.07793 [cs.DM], 2016.

%H Paul Barry, <a href="http://arxiv.org/abs/1105.3043">Eulerian polynomials as moments, via exponential Riordan arrays</a>, arXiv preprint arXiv:1105.3043 [math.CO], 2011, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry7/barry172.html">J. Int. Seq. 14 (2011) # 11.9.5</a>.

%H Paul Barry, <a href="http://arxiv.org/abs/1105.3044">Combinatorial polynomials as moments, Hankel transforms and exponential Riordan arrays</a>, arXiv preprint arXiv:1105.3044 [math.CO], 2011.

%H Paul Barry, <a href="https://arxiv.org/abs/1802.03443">On a transformation of Riordan moment sequences</a>, arXiv:1802.03443 [math.CO], 2018.

%H D. Barsky, <a href="http://www.mat.univie.ac.at/~slc/opapers/s05barsky.html">Analyse p-adique et suites classiques de nombres</a>, Sem. Loth. Comb. B05b (1981) 1-21.

%H H. Belbachir, M. Rahmani, B. Sury, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Rahmani/rahmani3.html">Sums Involving Moments of Reciprocals of Binomial Coefficients</a>, J. Int. Seq. 14 (2011) #11.6.6

%H Hacene Belbachir, Mourad Rahmani and B. Sury, <a href="http://www.emis.de/journals/JIS/VOL15/Sury/sury42.html">Alternating Sums of the Reciprocals of Binomial Coefficients</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.2.8.

%H F. Bergeron, Ph. Flajolet and B. Salvy, <a href="http://algo.inria.fr/flajolet/Publications/BeFlSa92.pdf">Varieties of Increasing Trees</a>, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.

%H V. Buchstaber and E. Bunkova, <a href="http://arxiv.org/abs/1010.0944">Elliptic formal group laws, integral Hirzebruch genera and Krichever genera</a>, arXiv preprint arXiv:1010.0944 [math-ph], 2010, p. 35.

%H F. Cachazo, S. He, E. Y. Yuan, <a href="http://arxiv.org/abs/1306.2962">Scattering in Three Dimensions from Rational Maps</a>, arXiv preprint arXiv:1306.2962 [hep-th], 2013.

%H F. Cachazo, S. Mizera, G. Zhang, <a href="http://arxiv.org/abs/1609.00008">Scattering equations: Real solutions and particles on a line</a>, arXiv preprint arXiv:1609.00008 [hep-th], 2016.

%H David Callan, Shi-Mei Ma, Toufik Mansour, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p22/">Some Combinatorial Arrays Related to the Lotka-Volterra System</a>, Electronic Journal of Combinatorics, Volume 22, Issue 2 (2015), Paper #P2.22.

%H Naiomi Cameron, J. E. McLeod, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/McLeod/mcleod3.html">Returns and Hills on Generalized Dyck Paths</a>, Journal of Integer Sequences, Vol. 19, 2016, #16.6.1.

%H L. Carlitz et al., <a href="http://dx.doi.org/10.1016/S0021-9800(66)80057-1">Permutations and sequences with repetitions by number of increases</a>, J. Combin. Theory, 1 (1966), 350-374, p. 351.

%H L. Carlitz, <a href="http://www.raco.cat/index.php/CollectaneaMathematica/article/view/57607/0">Eulerian numbers and operators</a>, Collectanea Mathematica 24:2 (1973), pp. 175-200.

%H Raphaël Cerf, Joseba Dalmau, <a href="https://arxiv.org/abs/1609.05738">The quasispecies distribution</a>, arXiv:1609.05738 [q-bio.PE], 2016.

%H Mircea I. Cirnu, <a href="http://www.emis.de/journals/BAMV/conten/vol18/BAMV_XVIII-1_p015-028.pdf">Determinantal formulas for sum of generalized arithmetic-geometric series</a>, Boletin de la Asociacion Matematica Venezolana, Vol. XVIII, No. 1 (2011), p. 13.

%H Tom Copeland, <a href="https://tcjpn.wordpress.com/2015/10/12/the-elliptic-lie-triad-kdv-and-ricattt-equations-infinigens-and-elliptic-genera/">The Elliptic Lie Triad: Riccati and KdV Equations, Infinigens, and Elliptic Genera</a>

%H J. Desarmenien and D. Foata, <a href="http://www-irma.u-strasbg.fr/~foata/paper/pub62.html">The signed Eulerian Numbers</a>

%H J. Desarmenien and D. Foata, <a href="http://dx.doi.org/10.1016/0012-365X(92)90364-L">The signed Eulerian numbers</a>, Discrete Math. 99 (1992), no. 1-3, 49-58.

%H E. Deutsch and B. E. Sagan, <a href="http://arxiv.org/abs/math.CO/0407326">Congruences for Catalan and Motzkin numbers and related sequences</a>, J. Num. Theory 117 (2006), 191-215.

%H D. Dominici, <a href="http://arxiv.org/abs/math/0501052">Nested derivatives: A simple method for computing series expansions of inverse functions.</a> arXiv:math/0501052v2 [math.CA]

%H B. Drake, <a href="http://people.brandeis.edu/~gessel/homepage/students/drakethesis.pdf">An inversion theorem for labeled trees and some limits of areas under lattice paths</a> Thesis, Brandeis Univ., Aug. 2008

%H C. Dupont, <a href="https://arxiv.org/abs/1601.00950">Odd zeta motive and linear forms in odd zeta values</a>, arXiv:1601.00950 [math.AG], 2016.

%H A. Dzhumadil'daev, D. Yeliussizov, <a href="http://cs.uwaterloo.ca/journals/JIS/VOL16/Yeliussizov/dzhuma6.html">Power sums of binomial coefficients</a>, Journal of Integer Sequences, 16 (2013), Article 13.1.6

%H R. Ehrenborg, M. Readdy, E. Steingrímsson, <a href="http://dx.doi.org/10.1006/jcta.1997.2832">Mixed Volumes and Slices of the Cube</a>, J Comb. Theory, Series A 81, Issue 1, Jan. 1998, 121-126.

%H M. Farber and A. Postnikov, <a href="http://arxiv.org/abs/1502.01434">Arrangements of equal minors in the positive Grassmannian</a>, arXiv preprint arXiv:1502.01434 [math.CO], 2015.

%H FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000021/">The number of descents of a permutation</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000155/">The number of exceedances (also excedences) of a permutation</a>, and <a href="http://www.findstat.org/StatisticsDatabase/St000213/">The number of weak exceedances (also weak excedences) of a permutation</a>

%H D. Foata, M. Schutzenberger, <a href="https://arxiv.org/abs/math/0508232">Théorie Géométrique des Polynômes Eulériens</a>, Lecture Notes in Math., no.138, Springer Verlag 1970; arXiv:math/0508232 [math.CO], 2005.

%H Dominique Foata and Guo-Niu Han, <a href="http://dx.doi.org/10.1093/qmath/hap043">Doubloons and new q-tangent numbers</a>, Quart. J. Math. 62 (2) (2011) 417-432

%H E. T. Frankel, <a href="/A000217/a000217_1.pdf"> A calculus of figurate numbers and finite differences</a>, American Mathematical Monthly, 57 (1950), 14-25. [Annotated scanned copy]

%H Ghislain R. Franssens, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Franssens/franssens13.html">On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.

%H S. Garoufalidis and R. Kashaev, <a href="http://arxiv.org/abs/1304.2705">From state integrals to q-series</a>, arXiv preprint arXiv:1304.2705 [math.GT], 2013.

%H Ira Gessel, <a href="http://www.cs.brandeis.edu/~ira/">The Smith College diploma problem</a>.

%H Alexander Gnedin, Grigori Olshanski, <a href="http://arxiv.org/abs/math/0602610">The boundary of the Eulerian number triangle</a>, arXiv:math/0602610 [math.PR], 2006.

%H Mats Granvik, <a href="http://math.stackexchange.com/questions/1085837/do-these-ratios-of-the-eulerian-number-triangle-converge-to-the-logarithm-of-x">Do these ratios of the Eulerian numbers converge to the logarithm of x?</a>, Math Stack Exchange, Dec 30 2014.

%H Jim Haglund and Mirko Visontai, <a href="http://hans.math.upenn.edu/~jhaglund/preprints/es-final.pdf">Stable multivariate Eulerian polynomials and generalized Stirling permutations</a>.

%H Thomas Hameister, Sujit Rao, Connor Simpson, <a href="http://www-users.math.umn.edu/~reiner/REU/HameisterRaoSimpson2017.pdf">Chow rings of matroids and atomistic lattices</a>, research paper, University of Minnesota, 2017, also <a href="https://arxiv.org/abs/1802.04241">arXiv:1802.04241</a> [math.CO].

%H Herwig Hauser, Christoph Koutschan, <a href="http://dx.doi.org/10.1016/j.disc.2012.08.009">Multivariate linear recurrences and power series division</a>, Discrete Math. 312 (2012), no. 24, 3553--3560. MR2979485.

%H F. Hirzebruch, <a href="http://wwwmath.uni-muenster.de/42/fileadmin/Einrichtungen/mjm/vol_1/mjm_vol_1_02.pdf">Eulerian polynomials</a>, Munster J. of Math. 1 (2008), pp. 9-12.

%H P. Hitczenko and S. Janson, <a href="http://arxiv.org/abs/1212.5498">Weighted random staircase tableaux</a>, arXiv preprint arXiv:1212.5498 [math.CO], 2012.

%H Matthew Hubbard and Tom Roby, <a href="http://web.archive.org/web/20080511174744/ http://binomial.csuhayward.edu/Euler.html">Pascal's Triangle From Top to Bottom</a>

%H Svante Janson, <a href="http://arxiv.org/abs/1305.3512">Euler-Frobenius numbers and rounding</a>, arXiv preprint arXiv:1305.3512 [math.PR], 2013.

%H Lucas Kang, <a href="http://arxiv.org/abs/1310.3311">Investigation of Rule 73 as Case Study of Class 4 Long-Distance Cellular Automata</a>, arXiv preprint arXiv:1310.3311 [nlin.CG], 2013.

%H A. Kerber and K.-J. Thuerlings, <a href="http://www.emis.de/journals/SLC/opapers/s08kerthur.html">Eulerian numbers, Foulkes characters and Lefschetz characters of S_n</a>, Séminaire Lotharingien, Vol. 8 (1984), 31-36.

%H A. R. Kräuter, <a href="http://www.mat.univie.ac.at/~slc/opapers/s11kraeu.html">Über die Permanente gewisser zirkulärer Matrizen...</a>, Séminaire Lotharingien de Combinatoire, B11b (1984), 11 pp.

%H D. H. Lehmer, <a href="http://dx.doi.org/10.1016/0097-3165(82)90020-6">Generalized Eulerian numbers</a>, J. Combin. Theory Ser.A 32 (1982), no. 2, 195--215. MR0654621 (83k:10026).

%H C. Lenart and K. Zainoulline, <a href="http://arxiv.org/abs/1408.5952">Towards generalized cohomology Schubert calculus via formal root polynomials</a>, arXiv preprint arXiv:1408.5952 [math.AG], 2014.

%H Nan Li, <a href="http://dx.doi.org/10.1007/s00454-012-9452-2">Ehrhart h*-vectors of hypersimplices</a>, Discr. Comp. Geom. 48 (2012) 847-878, Theorem 1.1

%H M-H. Li and N-C. Wong, <a href="http://www.math.nsysu.edu.tw/~wong/papers/soa-SEAM-formatted.pdf">Sums of angles of star polygons and the Eulerian Numbers</a>, Southeast Asian Bulletin of Mathematics 2004.

%H A. Losev and Y. Manin, <a href="http://arxiv.org/abs/math/0001003">New moduli spaces of pointed curves and pencils of flat connections</a>, arXiv preprint arXiv:0001003, 2000 (p. 8)

%H Shi-Mei Ma, <a href="http://arxiv.org/abs/1208.3104">Some combinatorial sequences associated with context-free grammars</a>, arXiv:1208.3104v2 [math.CO], 2012.

%H Shi-Mei Ma, <a href="http://arxiv.org/abs/1304.6654">On gamma-vectors and the derivatives of the tangent and secant functions</a>, arXiv:1304.6654 [math.CO], 2013.

%H Shi-Mei Ma, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i1p11">A family of two-variable derivative polynomials for tangent and secant</a>, El J. Combinat. 20 (1) (2013) P11

%H Shi-Mei Ma, Jun Ma, Yeong-Nan Yeh, <a href="https://arxiv.org/abs/1802.02861">On certain combinatorial expansions of descent polynomials and the change of grammars</a>, arXiv:1802.02861 [math.CO], 2018.

%H S.-M. Ma, T. Mansour, M. Schork, <a href="http://arxiv.org/abs/1308.0169">Normal ordering problem and the extensions of the Stirling grammar</a>, arXiv preprint arXiv:1308.0169 [math.CO], 2013.

%H Shi-Mei Ma, T. Mansour, D. Callan, <a href="http://arxiv.org/abs/1404.0731">Some combinatorial arrays related to the Lotka-Volterra system</a>, arXiv preprint arXiv:1404.0731 [math.CO], 2014.

%H Shi-Mei Ma, Hai-Na Wang, <a href="http://arxiv.org/abs/1506.08716">Enumeration of a dual set of Stirling permutations by their alternating runs</a>, arXiv:1506.08716 [math.CO], 2015.

%H P. A. MacMahon, <a href="https://doi.org/10.1112/plms/s2-19.1.305">The divisors of numbers</a>, Proc. London Math. Soc., (2) 19 (1920), 305-340; Coll. Papers II, pp. 267-302.

%H R. Mantaci and F. Rakotondrajao, <a href="http://dmtcs.episciences.org/271">A permutation representation that knows what "Eulerian" means</a>, Discrete Mathematics and Theoretical Computer Science, 4 101-108, (2001) [<a href="https://www.researchgate.net/publication/26645221">another version</a>]

%H O. J. Munch, <a href="/A000460/a000460.pdf">Om potensproduktsummer</a> [Norwegian, English summary], Nordisk Matematisk Tidskrift, 7 (1959), 5-19. [Annotated scanned copy]

%H O. J. Munch, <a href="http://www.jstor.org/stable/24524919">Om potensproduktsummer</a> [ Norwegian, English summary ], Nordisk Matematisk Tidskrift, 7 (1959), 5-19.

%H Nagatomo Nakamura, <a href="http://libir.josai.ac.jp/il/user_contents/02/G0000284repository/pdf/JOS-13447777-0808.pdf">Pseudo-Normal Random Number Generation via the Eulerian Numbers</a>, Josai Mathematical Monographs, vol 8, p 85-95, 2015.

%H S. Parker, <a href="http://people.brandeis.edu/~gessel/homepage/students/parkerthesis.pdf">The Combinatorics of Functional Composition and Inversion</a>, Dissertation, Brandeis Univ. (1993)

%H Vincent Pilaud, V Pons, <a href="http://arxiv.org/abs/1606.09643">Permutrees</a>, arXiv preprint arXiv:1606.09643, 2016

%H C. de Jesus Pita Ruiz Velasco, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Pita/pita5.html">Convolution and Sulanke Numbers</a>, JIS 13 (2010) 10.1.8

%H P. A. Piza, <a href="http://www.jstor.org/stable/3029339">Kummer numbers</a>, Mathematics Magazine, 21 (1947/1948), 257-260.

%H P. A. Piza, <a href="/A001117/a001117.pdf">Kummer numbers</a>, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy]

%H A. Randrianarivony and J. Zeng, <a href="http://dx.doi.org/10.1006/aama.1996.0001">Une famille de polynomes qui interpole plusieurs suites...</a>, Adv. Appl. Math. 17 (1996), 1-26.

%H J. Riordan, <a href="/A000217/a000217_2.pdf">Review of Frankel (1950)</a> [Annotated scanned copy]

%H J. Riordan, <a href="http://dx.doi.org/10.1090/S0002-9939-1951-0041090-7">Triangular permutation numbers</a>, Proc. Amer. Math. Soc. 2 (1951) 429-432, r(x,t).

%H D. P. Roselle, <a href="/A046739/a046739.pdf"> Permutations by number of rises and successions</a>, Proc. Amer. Math. Soc., 19 (1968), 8-16. [Annotated scanned copy]

%H G. Rzadkowski, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Rzadkowski/rzadkowski3.html">Two formulas for Successive Derivatives and Their Applications</a>, JIS 12 (2009) 09.8.2

%H Grzegorz Rzadkowski, M Urlinska, <a href="http://arxiv.org/abs/1612.06635">A Generalization of the Eulerian Numbers</a>, arXiv preprint arXiv:1612.06635, 2016

%H J. Sack and H. Ulfarsson, <a href="http://arxiv.org/abs/1106.1995">Refined inversion statistics on permutations</a>, arXiv preprint arXiv:1106.1995 [math.CO], 2011.

%H M. Sheppeard, <a href="http://vixra.org/pdf/1208.0242v6.pdf">Constructive motives and scattering</a> 2013 (p. 41).

%H D. Singh, <a href="/A002627/a002627.pdf">The numbers L(m,n) and their relations with prepared Bernoulli and Eulerian numbers</a>, Math. Student, 20 (1952), 66-70. [Annotated scanned copy]

%H M. Z. Spivey, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Spivey/spivey31.html">On Solutions to a General Combinatorial Recurrence</a>, J. Int. Seq. 14 (2011) # 11.9.7.

%H R. Sprugnoli, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Sprugnoli/sprugnoli6.html">Alternating Weighted Sums of Inverses of Binomial Coefficients</a>, J. Integer Sequences, 15 (2012), #12.6.3.

%H S. Tanimoto, <a href="http://dx.doi.org/10.1016/S0195-6698(02)00137-3">A study of Eulerian numbers by means of an operator on permutations</a>, Europ. J. Combin., 24 (2003), 33-43.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EulerianNumber.html">Eulerian Number</a> and <a href="http://mathworld.wolfram.com/EulersNumberTriangle.html">Euler's Number Triangle</a>

%H Susanne Wienand, <a href="https://oeis.org/wiki/File:Exceedances_4.png">plots of exceedances for permutations of [4]</a>

%H L. K. Williams, <a href="http://arXiv.org/abs/math.CO/0307271">Enumeration of totally positive Grassmann cells</a>, arXiv:math/0307271 [math.CO], 2003-2004.

%H Tingyao Xiong, Jonathan I. Hall, and Hung-Ping Tsao, <a href="http://dx.doi.org/10.1155/2014/870596">Combinatorial Interpretation of General Eulerian Numbers</a>, Journal of Discrete Mathematics, (2014), Article ID 870596, 6 pages.

%H D. Yeliussizov, <a href="http://web.archive.org/web/20160927104833/ http://www.kazntu.kz/sites/default/files/20121221ND_Eleusizov.pdf">Permutation Statistics on Multisets</a>, Ph.D. Dissertation, Computer Science, Kazakh-British Technical University, 2012

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%H <a href="/index/Cor#core">Index entries for "core" sequences</a>

%F T(n, k) = k * T(n-1, k) + (n-k+1) * T(n-1, k-1), T(1, 1) = 1.

%F T(n, k) = Sum_{j=0..k} (-1)^j * (k-j)^n * binomial(n+1, j).

%F Row sums = n! = A000142(n) unless n=0. - _Michael Somos_, Mar 17 2011

%F E.g.f. A(x, q) = Sum_{n>0} (Sum_{k=1..n} T(n, k) * q^k) * x^n / n! = q * ( e^(q*x) - e^x ) / ( q*e^x - e^(q*x) ) satisfies dA / dx = (A + 1) * (A + q). - _Michael Somos_, Mar 17 2011

%F For a column listing, n-th term: T(c, n) = c^(n+c-1) + Sum_{i=1..c-1} (-1)^i/i!*(c-i)^(n+c-1)*Product_{j=1..i} (n+c+1-j). - Randall L. Rathbun (randallr(AT)abac.com), Jan 23 2002

%F From John Robertson (jpr2718(AT)aol.com), Sep 02 2002: (Start)

%F Four characterizations of Eulerian numbers T(i, n):

%F 1. T(0, n)=1 for n>=1, T(i, 1)=0 for i>=1, T(i, n) = (n-i)T(i-1, n-1) + (i+1)T(i, n-1).

%F 2. T(i, n) = Sum_{j=0..i} (-1)^j*binomial(n+1,j)*(i-j+1)^n for n>=1, i>=0.

%F 3. Let C_n be the unit cube in R^n with vertices (e_1, e_2, ..., e_n) where each e_i is 0 or 1 and all 2^n combinations are used. Then T(i, n)/n! is the volume of C_n between the hyperplanes x_1 + x_2 + ... + x_n = i and x_1 + x_2 + ... + x_n = i+1. Hence T(i, n)/n! is the probability that i <= X_1 + X_2 + ... + X_n < i+1 where the X_j are independent uniform [0, 1] distributions. - See Ehrenborg & Readdy reference.

%F 4. Let f(i, n) = T(i, n)/n!. The f(i, n) are the unique coefficients so that (1/(r-1)^(n+1)) Sum_{i=0..n-1} f(i, n) r^{i+1} = Sum_{j>=0} (j^n)/(r^j) whenever n>=1 and abs(r)>1. (End)

%F O.g.f. for n-th row: (1-x)^(n+1)*polylog(-n, x)/x. - _Vladeta Jovovic_, Sep 02 2002

%F Triangle T(n, k), n>0 and k>0, read by rows; given by [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, ...] DELTA [1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, ...] (positive integers interspersed with 0's) where DELTA is Deléham's operator defined in A084938.

%F Sum_{k=1..n} T(n, k)*2^k = A000629(n). - _Philippe Deléham_, Jun 05 2004

%F From _Tom Copeland_, Oct 10 2007: (Start)

%F Bell(n,x) = Sum_{j=0..n) S2(n,j) * x^j = Sum_{j=0..n} E(n,j) * Lag(n,-x, j-n) = Sum_{j=0..n} (E(n,j)/n!) * (n!*Lag(n,-x, j-n)) = Sum_{j=0..n} E(n,j) * binomial(Bell(.,x)+j, n) umbrally where Bell(n,x) are the Bell / Touchard / exponential polynomials; S2(n,j), the Stirling numbers of the second kind; E(n,j), the Eulerian numbers; and Lag(n,x,m), the associated Laguerre polynomials of order m.

%F For x = 0, the equation gives Sum_{j=0..n} E(n,j) * binomial(j,n) = 1 for n=0 and 0 for all other n. By substituting the umbral compositional inverse of the Bell polynomials, the lower factorial n!*binomial(y,n), for x in the equation, the Worpitzky identity is obtained; y^n = Sum_{j=0..n} E(n,j) * binomial(y+j,n).

%F Note that E(n,j)/n! = E(n,j)/(Sum_{k=0..n}E(n,k)). Also (n!*Lag(n, -1, j-n)) is A086885 with a simple combinatorial interpretation in terms of seating arrangements, giving a combinatorial interpretation to the equation for x=1; n!*Bell(n,1) = n!*Sum_{j=0..n} S2(n,j) = Sum_{j=0..n} E(n,j) * (n!*Lag(n, -1, j-n)).

%F Clarification (Apr 19 2014): Here E(0,0) = S2(0,0) = 1, and for k>0, E(0,k) = E(k,0) = S2(0,k) = S2(k,0) = 0. (End)

%F From the relations between the h- and f-polynomials of permutohedra and reciprocals of e.g.f.s described in A049019: (t-1)((t-1)d/dx)^n 1/(t-exp(x)) evaluated at x=0 gives the n-th Eulerian row polynomial in t and the n-th row polynomial in (t-1) of A019538 and A090582. From the Comtet and Copeland references in A139605: ((t+exp(x)-1)d/dx)^(n+1) x gives pairs of the Eulerian polynomials in t as the coefficients of x^0 and x^1 in its Taylor series expansion in x. - _Tom Copeland_, Oct 05 2008

%F G.f: 1/(1-x/(1-x*y/1-2x/(1-2x*y/(1-3x/(1-3x*y/(1-... (continued fraction). - _Paul Barry_, Mar 24 2010

%F If n is odd prime, then the following consecutive 2*n+1 terms are 1 modulo n: a((n-1)*(n-2)/2+i), i=0,...,2*n. This chain of terms is maximal in the sense that neither the previous term nor the following one are 1 modulo n. - _Vladimir Shevelev, Jul 01 2011

%F From _Peter Bala_, Sep 29 2011: (Start)

%F For k = 0,1,2,... put G(k,x,t) := x -(1+2^k*t)*x^2/2 +(1+2^k*t+3^k*t^2)*x^3/3-(1+2^k*t+3^k*t^2+4^k*t^3)*x^4/4+.... Then the series reversion of G(k,x,t) with respect to x gives an e.g.f. for the present table when k = 0 and for A008517 when k = 1.

%F The e.g.f. B(x,t) := compositional inverse with respect to x of G(0,x,t) = (exp(x)-exp(x*t))/(exp(x*t)-t*exp(x)) = x + (1+t)*x^2/2! + (1+4*t+t^2)*x^3/3! + ... satisfies the autonomous differential equation dB/dx = (1+B)*(1+t*B) = 1 + (1+t)*B + t*B^2.

%F Applying [Bergeron et al., Theorem 1] gives a combinatorial interpretation for the Eulerian polynomials: A(n,t) counts plane increasing trees on n vertices where each vertex has outdegree <= 2, the vertices of outdegree 1 come in 1+t colors and the vertices of outdegree 2 come in t colors. An example is given below. Cf. A008517. Applying [Dominici, Theorem 4.1] gives the following method for calculating the Eulerian polynomials: Let f(x,t) = (1+x)*(1+t*x) and let D be the operator f(x,t)*d/dx. Then A(n+1,t) = D^n(f(x,t)) evaluated at x = 0.

%F (End)

%F With e.g.f. A(x,t)= G[x,(t-1)]-1 in Copeland's 2008 comment, the compositional inverse is Ainv(x,t)= log(t-(t-1)/(1+x))/(t-1). - _Tom Copeland_, Oct 11 2011

%F T(2*n+1,n+1) = (2*n+2)*T(2*n,n). (E.g., 66 = 6*11, 2416 = 8*302, ...) - _Gary Detlefs_, Nov 11 2011

%F E.g.f.: (1-y) / (1 - y*exp( (1-y)*x )). - _Geoffrey Critzer_, Nov 10 2012

%F From _Peter Bala_, Mar 12 2013: (Start)

%F Let {A(n,x)}n>=1 denote the sequence of Eulerian polynomials beginning [1, 1 + x, 1 + 4*x + x^2, ...]. Given two complex numbers a and b, the polynomial sequence defined by R(n,x) := (x+b)^n*A(n+1,(x+a)/(x+b)), n >= 0, satisfies the recurrence equation R(n+1,x) = d/dx((x+a)*(x+b)*R(n,x)). These polynomials give the row generating polynomials for several triangles in the database including A019538 (a = 0, b = 1), A156992 (a = 1, b = 1), A185421 (a = (1+i)/2, b = (1-i)/2), A185423 (a = exp(i*Pi/3), b = exp(-i*Pi/3)) and A185896 (a = i, b = -i).

%F (End)

%F E.g.f.: 1 + x/(T(0) - x*y), where T(k) = 1 + x*(y-1)/(1 + (k+1)/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Nov 07 2013

%F From _Tom Copeland_, Sep 18 2014: (Start)

%F A) Bivariate e.g.f. A(x,a,b)= (e^(ax)-e^(bx))/(a*e^(bx)-b*e^(ax)) = x + (a+b)*x^2/2! + (a^2+4ab+b^2)*x^3/3! + (a^3+11a^2b+11ab^2+b^3)x^4/4! + ...

%F B) B(x,a,b)= log((1+ax)/(1+bx))/(a-b) = x - (a+b)x^2/2 + (a^2+ab+b^2)x^3/3 - (a^3+a^2b+ab^2+b^3)x^4/4 + ... = log(1+u.*x), with (u.)^n = u_n = h_(n-1)(a,b) a complete homogeneous polynomial, is the compositional inverse of A(x,a,b) in x (see Drake, p. 56).

%F C) A(x) satisfies dA/dx = (1+a*A)(1+b*A) and can be written in terms of a Weierstrass elliptic function (see Buchstaber & Bunkova).

%F D) The bivariate Eulerian row polynomials are generated by the iterated derivative ((1+ax)(1+bx)d/dx)^n x evaluated at x=0 (see A145271).

%F E) A(x,a,b)= -(e^(-ax)-e^(-bx))/(a*e^(-ax)-b*e^(-bx)), A(x,-1,-1) = x/(1+x), and B(x,-1,-1) = x/(1-x).

%F F) FGL(x,y) = A(B(x,a,b) + B(y,a,b),a,b) = (x+y+(a+b)xy)/(1-ab*xy) is called the hyperbolic formal group law and related to a generalized cohomology theory by Lenart and Zainoulline. (End)

%F For x > 1, the n-th Eulerian polynomial A(n,x) = (x - 1)^n * log(x) * Integral_{u>=0} (ceiling(u))^n * x^(-u) du. - _Peter Bala_, Feb 06 2015

%F Sum_{j>=0} j^n/e^j, for n>=0, equals Sum_{k=1..n} T(n,k)e^k/(e-1)^(n+1), a rational function in the variable "e" which evaluates, approximately, to n! when e = A001113 = 2.71828... - _Richard R. Forberg_, Feb 15 2015

%F For a fixed k, T(n,k) ~ k^n, proved by induction. - _Ran Pan_, Oct 12 2015

%F From A145271, multiply the n-th diagonal (with n=0 the main diagonal) of the lower triangular Pascal matrix by g_n = (d/dx)^n (1+a*x)*(1+b*x) evaluated at x= 0, i.e., g_0 = 1, g_1 = (a+b), g_2 = 2ab, and g_n = 0 otherwise, to obtain the tridiagonal matrix VP with VP(n,k) = binomial(n,k) g_(n-k). Then the m-th bivariate row polynomial of this entry is P(m,a,b) = (1, 0, 0, 0,..) [VP * S]^(m-1) (1, a+b, 2ab, 0, ..)^T, where S is the shift matrix A129185, representing differentiation in the divided powers basis x^n/n!. Also, P(m,a,b) = (1, 0, 0, 0,..) [VP * S]^m (0, 1, 0, ..)^T. - _Tom Copeland_, Aug 02 2016

%e The triangle T(n, k) begins:

%e n\k 1 2 3 4 5 6 7 8 9 10 ...

%e 1: 1

%e 2: 1 1

%e 3: 1 4 1

%e 4: 1 11 11 1

%e 5: 1 26 66 26 1

%e 6: 1 57 302 302 57 1

%e 7: 1 120 1191 2416 1191 120 1

%e 8: 1 247 4293 15619 15619 4293 247 1

%e 9: 1 502 14608 88234 156190 88234 14608 502 1

%e 10: 1 1013 47840 455192 1310354 1310354 455192 47840 1013 1

%e ... Reformatted. - _Wolfdieter Lang_, Feb 14 2015

%e -----------------------------------------------------------------

%e E.g.f. = (y) * x^1 / 1! + (y + y^2) * x^2 / 2! + (y + 4*y^2 + y^3) * x^3 / 3! + ... - _Michael Somos_, Mar 17 2011

%e Let n=7. Then the following 2*7+1=15 consecutive terms are 1(mod 7): a(15+i), i=0..14. - _Vladimir Shevelev_, Jul 01 2011

%e Row 3: The plane increasing 0-1-2 trees on 3 vertices (with the number of colored vertices shown to the right of a vertex) are

%e .

%e . 1o (1+t) 1o t 1o t

%e . | / \ / \

%e . | / \ / \

%e . 2o (1+t) 2o 3o 3o 2o

%e . |

%e . |

%e . 3o

%e .

%e The total number of trees is (1+t)^2 + t + t = 1 + 4*t + t^2.

%p A008292 := proc(n,k) option remember; if k < 1 or k > n then 0; elif k = 1 or k = n then 1; else k*procname(n-1,k)+(n-k+1)*procname(n-1,k-1) ; end if; end proc:

%t len = 54; m = Ceiling[2 Sqrt[len]]; t[n_, k_] = Sum[(-1)^j*(k - j)^n*Binomial[n + 1, j], {j, 0, k}]; Flatten[Table[t[n, k], {n, 1, m}, {k, 1, n}]][[1 ;; len]] (* _Jean-François Alcover_, May 31 2011, after _Michael Somos_ *)

%t Flatten[Table[CoefficientList[(1-x)^(k+1)*PolyLog[-k, x]/x, x], {k, 1, 10}]] (* _Vaclav Kotesovec_, Aug 27 2015 *)

%o (PARI) {T(n, k) = if( k<1 || k>n, 0, if( n==1, 1, k * T(n-1, k) + (n-k+1) * T(n-1, k-1)))}; /* _Michael Somos_, Jul 19 1999 */

%o (PARI) {T(n, k) = sum( j=0, k, (-1)^j * (k-j)^n * binomial( n+1, j))}; /* _Michael Somos_, Jul 19 1999 */

%o {A008292(c,n)=c^(n+c-1)+sum(i=1,c-1,(-1)^i/i!*(c-i)^(n+c-1)*prod(j=1,i,n+c+1-j))}

%o (Haskell)

%o import Data.List (genericLength)

%o a008292 n k = a008292_tabl !! (n-1) !! (k-1)

%o a008292_row n = a008292_tabl !! (n-1)

%o a008292_tabl = iterate f [1] where

%o f xs = zipWith (+)

%o (zipWith (*) ([0] ++ xs) (reverse ks)) (zipWith (*) (xs ++ [0]) ks)

%o where ks = [1 .. 1 + genericLength xs]

%o -- _Reinhard Zumkeller_, May 07 2013

%o (Python)

%o from sympy import binomial

%o def T(n, k): return sum([(-1)**j*(k - j)**n*binomial(n + 1, j) for j in range(k + 1)])

%o for n in range(1, 11): print([T(n, k) for k in range(1, n + 1)]) # _Indranil Ghosh_, Nov 08 2017

%o (R)

%o T <- function(n, k) {

%o S <- numeric()

%o for (j in 0:k) S <- c(S, (-1)^j*(k-j)^n*choose(n+1, j))

%o return(sum(S))

%o }

%o for (n in 1:10){

%o for (k in 1:n) print(T(n,k))

%o } # _Indranil Ghosh_, Nov 08 2017

%K nonn,tabl,nice,eigen,core,look

%O 1,5

%A _N. J. A. Sloane_, Mar 15 1996

%E Thanks to _Michael Somos_ for additional comments.

%E Further comments from _Christian G. Bower_, May 12 2000

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified May 23 18:26 EDT 2018. Contains 304483 sequences. (Running on oeis4.)