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 A002426 M2673 N1070 #561 Jul 03 2025 15:26:00
%S A002426 1,1,3,7,19,51,141,393,1107,3139,8953,25653,73789,212941,616227,
%T A002426 1787607,5196627,15134931,44152809,128996853,377379369,1105350729,
%U A002426 3241135527,9513228123,27948336381,82176836301,241813226151,712070156203,2098240353907,6186675630819
%N A002426 Central trinomial coefficients: largest coefficient of (1 + x + x^2)^n.
%C A002426 Number of ordered trees with n + 1 edges, having root of odd degree and nonroot nodes of outdegree at most 2. - _Emeric Deutsch_, Aug 02 2002
%C A002426 Number of paths of length n with steps U = (1,1), D = (1,-1) and H = (1,0), running from (0,0) to (n,0) (i.e., grand Motzkin paths of length n). For example, a(3) = 7 because we have HHH, HUD, HDU, UDH, DUH, UHD and DHU. - _Emeric Deutsch_, May 31 2003
%C A002426 Number of lattice paths from (0,0) to (n,n) using steps (2,0), (0,2), (1,1). It appears that 1/sqrt((1 - x)^2 - 4*x^s) is the g.f. for lattice paths from (0,0) to (n,n) using steps (s,0), (0,s), (1,1). - _Joerg Arndt_, Jul 01 2011
%C A002426 Number of lattice paths from (0,0) to (n,n) using steps (1,0), (1,1), (1,2). - _Joerg Arndt_, Jul 05 2011
%C A002426 Binomial transform of A000984, with interpolated zeros. - _Paul Barry_, Jul 01 2003
%C A002426 Number of leaves in all 0-1-2 trees with n edges, n > 0. (A 0-1-2 tree is an ordered tree in which every vertex has at most two children.) - _Emeric Deutsch_, Nov 30 2003
%C A002426 a(n) is the number of UDU-free paths of n + 1 upsteps (U) and n downsteps (D) that start U. For example, a(2) = 3 counts UUUDD, UUDDU, UDDUU. - _David Callan_, Aug 18 2004
%C A002426 Diagonal sums of triangle A063007. - _Paul Barry_, Aug 31 2004
%C A002426 Number of ordered ballots from n voters that result in an equal number of votes for candidates A and B in a three candidate election. Ties are counted even when candidates A and B lose the election. For example, a(3) = 7 because ballots of the form (voter-1 choice, voter-2 choice, voter-3 choice) that result in equal votes for candidates A and B are the following: (A,B,C), (A,C,B), (B,A,C), (B,C,A), (C,A,B), (C,B,A) and (C,C,C). - _Dennis P. Walsh_, Oct 08 2004
%C A002426 a(n) is the number of weakly increasing sequences (a_1,a_2,...,a_n) with each a_i in [n]={1,2,...,n} and no element of [n] occurring more than twice. For n = 3, the sequences are 112, 113, 122, 123, 133, 223, 233. - _David Callan_, Oct 24 2004
%C A002426 Note that n divides a(n+1) - a(n). In fact, (a(n+1) - a(n))/n = A007971(n+1). - _T. D. Noe_, Mar 16 2005
%C A002426 Row sums of triangle A105868. - _Paul Barry_, Apr 23 2005
%C A002426 Number of paths of length n with steps U = (1,1), D = (1,-1) and H = (1,0), starting at (0,0), staying weakly above the x-axis (i.e., left factors of Motzkin paths) and having no H steps on the x-axis. Example: a(3) = 7 because we have UDU, UHD, UHH, UHU, UUD, UUH and UUU. - _Emeric Deutsch_, Oct 07 2007
%C A002426 Equals right border of triangle A152227; starting with offset 1, the row sums of triangle A152227. - _Gary W. Adamson_, Nov 29 2008
%C A002426 Starting with offset 1 = iterates of M * [1,1,1,...] where M = a tridiagonal matrix with [0,1,1,1,...] in the main diagonal and [1,1,1,...] in the super and subdiagonals. - _Gary W. Adamson_, Jan 07 2009
%C A002426 Hankel transform is 2^n. - _Paul Barry_, Aug 05 2009
%C A002426 a(n) is prime for n = 2, 3 and 4, with no others for n <= 10^5 (E. W. Weisstein, Mar 14 2005). It has apparently not been proved that no [other] prime central trinomials exist. - _Jonathan Vos Post_, Mar 19 2010
%C A002426 a(n) is not divisible by 3 for n whose base-3 representation contains no 2 (A005836).
%C A002426 a(n) = number of (n-1)-lettered words in the alphabet {1,2,3} with as many occurrences of the substring (consecutive subword) [1,2] as those of [2,1]. See the papers by Ekhad-Zeilberger and Zeilberger. - _N. J. A. Sloane_, Jul 05 2012
%C A002426 a(n) = coefficient of x^n in (1 + x + x^2)^n. - _L. Edson Jeffery_, Mar 23 2013
%C A002426 a(n) is the number of ordered pairs (A,B) of subsets of {1,2,...,n} such that (i.) A and B are disjoint and (ii.) A and B contain the same number of elements. For example, a(2) = 3 because we have: ({},{}) ; ({1},{2}) ; ({2},{1}). - _Geoffrey Critzer_, Sep 04 2013
%C A002426 Also central terms of A082601. - _Reinhard Zumkeller_, Apr 13 2014
%C A002426 a(n) is the number of n-tuples with entries 0, 1, or 2 and with the sum of entries equal to n. For n=3, the seven 3-tuples are (1,1,1), (0,1,2), (0,2,1), (1,0,2), (1,2,0), (2,0,1), and (2,1,0). - _Dennis P. Walsh_, May 08 2015
%C A002426 The series 2*a(n) + 3*a(n+1) + a(n+2) = 2*A245455(n+3) has Hankel transform of L(2n+1)*2^n, offset n = 1, L being a Lucas number, see A002878 (empirical observation). - _Tony Foster III_, Sep 05 2016
%C A002426 The series (2*a(n) + 3*a(n+1) + a(n+2))/2 = A245455(n+3) has Hankel transform of L(2n+1), offset n=1, L being a Lucas number, see A002878 (empirical observation). - _Tony Foster III_, Sep 05 2016
%C A002426 Conjecture: An integer n > 3 is prime if and only if a(n) == 1 (mod n^2). We have verified this for n up to 8*10^5, and proved that a(p) == 1 (mod p^2) for any prime p > 3 (cf. A277640). - _Zhi-Wei Sun_, Nov 30 2016
%C A002426 This is the analog for Coxeter type B of Motzkin numbers (A001006) for Coxeter type A. - _F. Chapoton_, Jul 19 2017
%C A002426 a(n) is also the number of solutions to the equation x(1) + x(2) + ... + x(n) = 0, where x(1), ..., x(n) are in the set {-1,0,1}. Indeed, the terms in (1 + x + x^2)^n that produce x^n are of the form x^i(1)*x^i(2)*...*x^i(n) where i(1), i(2), ..., i(n) are in {0,1,2} and i(1) + i(2) + ... + i(n) = n. By setting j(t) = i(t) - 1 we obtain that j(1), ..., j(n) satisfy j(1) + ... + j(n) =0 and j(t) in {-1,0,1} for all t = 1..n. - _Lucien Haddad_, Mar 10 2018
%C A002426 If n is a prime greater than 3 then a(n)-1 is divisible by n^2. - _Ira M. Gessel_, Aug 08 2021
%C A002426 Let f(m) = ceiling((q+log(q))/log(9)), where q = -log(log(27)/(2*m^2*Pi)) then f(a(n)) = n, for n > 0. - _Miko Labalan_, Oct 07 2024
%C A002426 Diagonal of the rational function 1 / (1 - x^2 - y^2 - x*y). - _Ilya Gutkovskiy_, Apr 23 2025
%D A002426 L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 78 and 163, #19.
%D A002426 L. Euler, Exemplum Memorabile Inductionis Fallacis, Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 15, p. 59.
%D A002426 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 575.
%D A002426 P. Henrici, Applied and Computational Complex Analysis. Wiley, NY, 3 vols., 1974-1986. (Vol. 1, p. 42.)
%D A002426 Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 579.
%D A002426 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 74.
%D A002426 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002426 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A002426 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 6.3.8.
%D A002426 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 22.
%D A002426 Lin Yang and S.-L. Yang, The parametric Pascal rhombus. Fib. Q., 57:4 (2019), 337-346. See p. 341.
%H A002426 Seiichi Manyama, <a href="/A002426/b002426.txt">Table of n, a(n) for n = 0..1000</a> (first 201 terms from T. D. Noe)
%H A002426 Katharine A. Ahrens, <a href="https://www.lib.ncsu.edu/resolver/1840.20/37364">Combinatorial Applications of the k-Fibonacci Numbers: A Cryptographically Motivated Analysis</a>, Ph. D. thesis, North Carolina State University (2020).
%H A002426 George E. Andrews, <a href="http://www.mat.univie.ac.at/~slc/opapers/s25andrews.html">Three aspects of partitions</a>, Séminaire Lotharingien de Combinatoire, B25f (1990), 1 p.
%H A002426 George E. Andrews, <a href="http://dx.doi.org/10.1090/S0894-0347-1990-1040390-4">Euler's 'exemplum memorabile inductionis fallacis' and q-trinomial coefficients</a>, J. Amer. Math. Soc. 3 (1990) 653-669.
%H A002426 Armen G. Bagdasaryan and Ovidiu Bagdasar, <a href="https://doi.org/10.1016/j.endm.2018.05.012">On some results concerning generalized arithmetic triangles</a>, Electronic Notes in Discrete Mathematics (2018) Vol. 67, 71-77.
%H A002426 E. Barcucci, R. Pinzani and R. Sprugnoli, <a href="https://users.dimi.uniud.it/~giacomo.dellariccia/Table%20of%20contents/BarcucciPinzaniSprugnoli1991.pdf">The Motzkin family</a>, P.U.M.A. Ser. A, Vol. 2, 1991, No. 3-4, pp. 249-279.
%H A002426 Paul Barry, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Barry/barry84.html">A Catalan Transform and Related Transformations on Integer Sequences</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%H A002426 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Barry3/barry93.html">Continued fractions and transformations of integer sequences</a>, JIS 12 (2009) 09.7.6.
%H A002426 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Barry/barry321.html">Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices</a>, Journal of Integer Sequences, 19, 2016, #16.3.5.
%H A002426 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Barry/barry444.html">On the Central Antecedents of Integer (and Other) Sequences</a>, J. Int. Seq., Vol. 23 (2020), Article 20.8.3.
%H A002426 Frank R. Bernhart, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00054-0">Catalan, Motzkin and Riordan numbers</a>, Discr. Math., 204 (1999) 73-112.
%H A002426 N. M. Bogoliubov, <a href="ftp://ftp.pdmi.ras.ru/pub/publicat/znsl/v487/p053.pdf">Enumerative combinatorics of XX0 Heisenberg chain</a>, Scientific Notes, POMI Workshops, Russian Academy of Sciences (St. Petersburg, Russia, 2019), Vol. 487.
%H A002426 Jan Bok, <a href="https://arxiv.org/abs/1801.05498">Graph-indexed random walks on special classes of graphs</a>, arXiv:1801.05498 [math.CO], 2018.
%H A002426 Johann Cigler, <a href="http://arxiv.org/abs/1109.1449">Some nice Hankel determinants</a>. arXiv preprint arXiv:1109.1449 [math.CO], 2011.
%H A002426 Johann Cigler and Christian Krattenthaler, <a href="https://arxiv.org/abs/2003.01676">Hankel determinants of linear combinations of moments of orthogonal polynomials</a>, arXiv:2003.01676 [math.CO], 2020.
%H A002426 Isaac DeJager, Madeleine Naquin and Frank Seidl, <a href="https://www.valpo.edu/mathematics-statistics/files/2019/08/Drube2019.pdf">Colored Motzkin Paths of Higher Order</a>, VERUM 2019.
%H A002426 Emeric Deutsch and B. E. Sagan, <a href="https://arxiv.org/abs/math/0407326">Congruences for Catalan and Motzkin numbers and related sequences</a>, arXiv:math/0407326 [math.CO], 2004; J. Num. Theory 117 (2006), 191-215.
%H A002426 Steffen Eger, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Eger/eger6.html">Restricted Weighted Integer Compositions and Extended Binomial Coefficients</a>, J. Integer. Seq., Vol. 16 (2013), #13.1.3. - From _N. J. A. Sloane_, Feb 03 2013
%H A002426 Steffen Eger, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.121.04.344">Stirling's Approximation for Central Extended Binomial Coefficients</a>, American Mathematical Monthly, 121 (2014), 344-349.
%H A002426 Shalosh B. Ekhad and Doron Zeilberger, <a href="http://arxiv.org/abs/1112.6207">Automatic Solution of Richard Stanley's Amer. Math. Monthly Problem #11610 and ANY Problem of That Type</a>, arXiv preprint arXiv:1112.6207 [math.CO], 2011.
%H A002426 Luca Ferrari and Emanuele Munarini, <a href="http://arxiv.org/abs/1203.6792">Enumeration of edges in some lattices of paths</a>, arXiv preprint arXiv:1203.6792 [math.CO], 2012 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Ferrari/ferrari.html">J. Int. Seq. 17 (2014) #14.1.5</a>
%H A002426 Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, <a href="http://arxiv.org/abs/1110.6638">Sato-Tate distributions and Galois endomorphism modules in genus 2</a>, arXiv preprint arXiv:1110.6638 [math.NT], 2011.
%H A002426 Francesc Fite and Andrew V. Sutherland, <a href="http://arxiv.org/abs/1203.1476">Sato-Tate distributions of twists of y^2=x^5-x and y^2=x^6+1</a>, arXiv preprint arXiv:1203.1476 [math.NT], 2012. - From _N. J. A. Sloane_, Sep 14 2012
%H A002426 Rigoberto Flórez, Leandro Junes and José L. Ramírez, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Florez/florez4.html">Further Results on Paths in an n-Dimensional Cubic Lattice</a>, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.2.
%H A002426 R. K. Guy, editor, <a href="/A339579/a339579.pdf">Western Number Theory Problems, 1985-12-21 & 23</a>, Typescript, Jul 13 1986, Dept. of Math. and Stat., Univ. Calgary, 11 pages. Annotated scan of pages 1, 3, 7, 9, with permission. See Problem 85:03.
%H A002426 R. K. Guy, <a href="/A005712/a005712.pdf">Letter to N. J. A. Sloane, 1987</a>
%H A002426 R. K. Guy, <a href="http://www.jstor.org/stable/2691503">The Second Strong Law of Small Numbers</a>, Math. Mag, 63 (1990) 3-20, esp. 18-19.
%H A002426 R. K. Guy, <a href="/A005347/a005347.pdf">The Second Strong Law of Small Numbers</a>, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
%H A002426 V. E. Hoggatt, Jr. and M. Bicknell, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/7-4/hoggatt-a.pdf">Diagonal sums of generalized Pascal triangles</a>, Fib. Quart., 7 (1969), 341-358, 393.
%H A002426 Po-Yi Huang, Shu-Chung Liu, and Yeong-Nan Yeh, <a href="https://doi.org/10.37236/3693">Congruences of Finite Summations of the Coefficients in certain Generating Functions</a>, The Electronic Journal of Combinatorics, 21 (2014), #P2.45.
%H A002426 Cynthia Huffman, <a href="https://doi.org/10.56031/2693-9908.1048">Analytical Observations (Translation of E326)</a>, Euleriana (2023) Vol. 3, Issue 1.
%H A002426 Anders Hyllengren, <a href="/A001006/a001006_5.pdf">Four integer sequences</a>, Oct 04 1985. Observes essentially that A000984 and A002426 are inverse binomial transforms of each other, as are A000108 and A001006.
%H A002426 Veronika Irvine, Stephen Melczer and Frank Ruskey, <a href="https://arxiv.org/abs/1804.08725">Vertically constrained Motzkin-like paths inspired by bobbin lace</a>, arXiv:1804.08725 [math.CO], 2018.
%H A002426 L. Kleinrock, <a href="/A027907/a027907.pdf">Uniform permutation of sequences</a>, JPL Space Programs Summary, Vol. 37-64-III, Apr 30, 1970, pp. 32-43. [Annotated scanned copy]
%H A002426 Nadav Kohen, <a href="https://arxiv.org/abs/2411.03681">Density and Symmetry in the Generalized Motzkin Numbers mod p</a>, arXiv:2411.03681 [math.CO], 2024. See p. 2.
%H A002426 Dmitry Kruchinin and Vladimir Kruchinin, <a href="http://cs.uwaterloo.ca/journals/JIS/VOL18/Kruchinin/kruch9.html">A Generating Function for the Diagonal T2n,n in Triangles</a>, Journal of Integer Sequence, Vol. 18 (2015), article 15.4.6.
%H A002426 Shara Lalo and Zagros Lalo, <a href="/A002426/a002426_3.pdf">Formula for the Central terms in triangle A027907 ((1 + x + x^2)^n)</a>.
%H A002426 John W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/LAYMAN/hankel.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%H A002426 Andrew Lohr, <a href="https://arxiv.org/abs/1805.00076">Several Topics in Experimental Mathematics</a>, arXiv:1805.00076 [math.CO], 2018.
%H A002426 Toufik Mansour and Mark Shattuck, <a href="https://math.colgate.edu/~integers/z5/z5.pdf">Enumeration of Catalan and smooth words according to capacity</a>, Integers (2025) Vol. 25, Art. No. A5. See pp. 28, 32.
%H A002426 Romeo Meštrović, <a href="http://arxiv.org/abs/1409.3820">Lucas' theorem: its generalizations, extensions and applications (1878--2014)</a>, arXiv preprint arXiv:1409.3820 [math.NT], 2014.
%H A002426 Thorsten Neuschel, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Neuschel/neuschel4.html">A Note on Extended Binomial Coefficients</a>, J. Int. Seq. 17 (2014) # 14.10.4.
%H A002426 Tony D. Noe, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Noe/noe35.html">On the Divisibility of Generalized Central Trinomial Coefficients</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
%H A002426 Paul Peart and Wen-Jin Woan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/PEART/peart1.html">Generating Functions via Hankel and Stieltjes Matrices</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.1.
%H A002426 Ed. Pegg, Jr., <a href="http://www.mathpuzzle.com/coin.html">Number of combinations of n coins when have 3 kinds of coin</a>
%H A002426 E. Pergola, R. Pinzani, S. Rinaldi and R. A. Sulanke, <a href="http://dx.doi.org/10.1006/aama.2001.0796">A bijective approach to the area of generalized Motzkin paths</a>, Adv. Appl. Math., 28, 2002, 580-591.
%H A002426 José L. Ramírez, <a href="http://arxiv.org/abs/1511.04577">The Pascal Rhombus and the Generalized Grand Motzkin Paths</a>, arXiv:1511.04577 [math.CO], 2015.
%H A002426 José L. Ramírez and Víctor F. Sirvent, <a href="https://doi.org/10.37236/4618">A Generalization of the k-Bonacci Sequence from Riordan Arrays</a>, The Electronic Journal of Combinatorics, 22(1) (2015), #P1.38.
%H A002426 Dan Romik, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL6/Romik/romik5.html">Some formulas for the central trinomial and Motzkin numbers</a>, J. Integer Seqs., Vol. 6, 2003.
%H A002426 Eric Rowland and Reem Yassawi, <a href="http://arxiv.org/abs/1310.8635">Automatic congruences for diagonals of rational functions</a>, arXiv preprint arXiv:1310.8635 [math.NT], 2013.
%H A002426 Michelle Rudolph-Lilith and Lyle E. Muller, <a href="http://arxiv.org/abs/1403.5942">On an explicit representation of central (2k+1)-nomial coefficients</a>, arXiv preprint arXiv:1403.5942 [math.CO], 2014.
%H A002426 Michelle Rudolph-Lilith and Lyle E. Muller, <a href="http://dx.doi.org/10.1006/aama.2001.0796">On a link between Dirichlet kernels and central multinomial coefficients</a>, Discrete Mathematics, Volume 338, Issue 9, Sep 06 2015, Pages 1567-1572.
%H A002426 Jesus Salas and Alan D. Sokal, <a href="http://arxiv.org/abs/0711.1738">Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. V. Further Results for the Square-Lattice Chromatic Polynomial</a>, arXiv:0711.1738 [cond-mat.stat-mech], 2007-2009; J. Stat. Phys. 135 (2009) 279-373, arXiv:0711.1738 [cond-mat.stat-mech]. Mentions this sequence.
%H A002426 Louis W. Shapiro, Seyoum Getu, Wen-Jin Woan, and Leon C. Woodson, <a href="http://dx.doi.org/10.1016/0166-218X(91)90088-E">The Riordan group</a>, Discrete Applied Math., 34 (1991), 229-239.
%H A002426 T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/series008">Middle Trinomial Coefficient</a>
%H A002426 Michael Z. Spivey and Laura L. Steil, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
%H A002426 Robert A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/SULANKE/sulanke.html">Moments of generalized Motzkin paths</a>, J. Integer Sequences, Vol. 3 (2000), #00.1.
%H A002426 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1208.2683">Conjectures involving combinatorial sequences</a>, arXiv preprint arXiv:1208.2683 [math.CO], 2012. - _N. J. A. Sloane_, Dec 25 2012
%H A002426 Zhi-Wei Sun, <a href="http://math.nju.edu.cn/~zwsun/142p.pdf">Conjectures involving arithmetical sequences</a>, Number Theory: Arithmetic in Shangri-La (eds., S. Kanemitsu, H.-Z. Li and J.-Y. Liu), Proc. the 6th China-Japan Sem. Number Theory (Shanghai, August 15-17, 2011), World Sci., Singapore, 2013, pp. 244-258. - _N. J. A. Sloane_, Dec 28 2012
%H A002426 Zhi-Wei Sun, <a href="https://mathoverflow.net/questions/491563">On central trinomial coefficients</a>, Question 491563 at MathOverflow, April 23, 2025.
%H A002426 Paveł Szabłowski, <a href="https://cdm.ucalgary.ca/article/view/76214">Beta distributions whose moment sequences are related to integer sequences listed in the OEIS</a>, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 96.
%H A002426 Dennis P. Walsh, <a href="http://www.mtsu.edu/~dwalsh/3votetie.gif">The Probability of a Tie in a Three Candidate Election</a>.
%H A002426 Yi Wang and Bao-Xuan Zhu, <a href="http://arxiv.org/abs/1303.5595">Proofs of some conjectures on monotonicity of number-theoretic and combinatorial sequences</a>, arXiv preprint arXiv:1303.5595 [math.CO], 2013.
%H A002426 Chenying Wang, Piotr Miska, and István Mező, <a href="http://doi.org/10.1016/j.disc.2016.10.012">The r-derangement numbers</a>, Discrete Mathematics 340.7 (2017): 1681-1692.
%H A002426 Chen Wang, <a href="https://arxiv.org/abs/2003.09888">Supercongruences and hypergeometric transformations</a>, arXiv:2003.09888 [math.NT], 2020.
%H A002426 Chen Wang and Zhi-Wei Sun, <a href="https://arxiv.org/abs/1910.06850">Congruences involving central trinomial coefficients</a>, arXiv:1910.06850 [math.NT], 2019.
%H A002426 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CentralTrinomialCoefficient.html">Central Trinomial Coefficient</a> and <a href="https://mathworld.wolfram.com/TrinomialCoefficient.html">Trinomial Coefficient</a>.
%H A002426 Doron Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/tokhniot/oRPS32">Analogs of the Richard Stanley Amer. Math. Monthly Problem 11610 for ALL pairs of words of length, 2, in an alphabet of, 3 letters.</a> See Proposition 5.
%H A002426 Doron Zeilberger, <a href="/A002426/a002426.txt">Analogs of the Richard Stanley Amer. Math. Monthly Problem 11610 for ALL pairs of words of length, 2, in an alphabet of, 3 letters.</a> [Local copy]
%H A002426 <a href="/index/Ce#cfn">Index entries for sequences of k-nomial coefficients</a>
%H A002426 <a href="/index/Cor#core">Index entries for "core" sequences</a>
%F A002426 G.f.: 1/sqrt(1 - 2*x - 3*x^2).
%F A002426 E.g.f.: exp(x)*I_0(2x), where I_0 is a Bessel function. - _Michael Somos_, Sep 09 2002
%F A002426 a(n) = 2*A027914(n) - 3^n. - _Benoit Cloitre_, Sep 28 2002
%F A002426 a(n) is asymptotic to d*3^n/sqrt(n) with d around 0.5.. - _Benoit Cloitre_, Nov 02 2002, d = sqrt(3/Pi)/2 = 0.4886025119... - Alec Mihailovs (alec(AT)mihailovs.com), Feb 24 2005
%F A002426 D-finite with recurrence: a(n) = ((2*n - 1)*a(n-1) + 3*(n - 1)*a(n-2))/n; a(0) = a(1) = 1; see paper by Barcucci, Pinzani and Sprugnoli.
%F A002426 Inverse binomial transform of A000984. - _Vladeta Jovovic_, Apr 28 2003
%F A002426 a(n) = Sum_{k=0..n} binomial(n, k)*binomial(k, k/2)*(1 + (-1)^k)/2; a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*binomial(2*k, k). - _Paul Barry_, Jul 01 2003
%F A002426 a(n) = Sum_{k>=0} binomial(n, 2*k)*binomial(2*k, k). - _Philippe Deléham_, Dec 31 2003
%F A002426 a(n) = Sum_{i+j=n, 0<=j<=i<=n} binomial(n, i)*binomial(i, j). - _Benoit Cloitre_, Jun 06 2004
%F A002426 a(n) = 3*a(n-1) - 2*A005043(n). - Joost Vermeij (joost_vermeij(AT)hotmail.com), Feb 10 2005
%F A002426 a(n) = Sum_{k=0..n} binomial(n, k)*binomial(k, n-k). - _Paul Barry_, Apr 23 2005
%F A002426 a(n) = (-1/4)^n*Sum_{k=0..n} binomial(2*k, k)*binomial(2*n-2*k, n-k)*(-3)^k. - _Philippe Deléham_, Aug 17 2005
%F A002426 a(n) = A111808(n,n). - _Reinhard Zumkeller_, Aug 17 2005
%F A002426 a(n) = Sum_{k=0..n} (((1 + (-1)^k)/2)*Sum_{i=0..floor((n-k)/2)} binomial(n, i)*binomial(n-i, i+k)*((k + 1)/(i + k + 1))). - _Paul Barry_, Sep 23 2005
%F A002426 a(n) = 3^n*Sum_{j=0..n} (-1/3)^j*C(n, j)*C(2*j, j); follows from (a) in A027907. - Loic Turban (turban(AT)lpm.u-nancy.fr), Aug 31 2006
%F A002426 a(n) = (1/2)^n*Sum_{j=0..n} 3^j*binomial(n, j)*binomial(2*n-2*j, n) = (3/2)^n*Sum_{j=0..n} (1/3)^j*binomial(n, j)*binomial(2*j, n); follows from (c) in A027907. - Loic Turban (turban(AT)lpm.u-nancy.fr), Aug 31 2006
%F A002426 a(n) = (1/Pi)*Integral_{x=-1..3} x^n/sqrt((3 - x)*(1 + x)) is moment representation. - _Paul Barry_, Sep 10 2007
%F A002426 G.f.: 1/(1 - x - 2x^2/(1 - x - x^2/(1 - x - x^2/(1 - ... (continued fraction). - _Paul Barry_, Aug 05 2009
%F A002426 a(n) = sqrt(-1/3)*(-1)^n*hypergeometric([1/2, n+1], [1], 4/3). - _Mark van Hoeij_, Nov 12 2009
%F A002426 a(n) = (1/Pi)*Integral_{x=-1..1} (1 + 2*x)^n/sqrt(1 - x^2) = (1/Pi)*Integral_{t=0..Pi} (1 + 2*cos(t))^n. - _Eli Wolfhagen_, Feb 01 2011
%F A002426 In general, g.f.: 1/sqrt(1 - 2*a*x + x^2*(a^2 - 4*b)) = 1/(1 - a*x)*(1 - 2*x^2*b/(G(0)*(a*x - 1) + 2*x^2*b)); G(k) = 1 - a*x - x^2*b/G(k+1); for g.f.: 1/sqrt(1 - 2*x - 3*x^2) = 1/(1 - x)*(1 - 2*x^2/(G(0)*(x - 1) + 2*x^2)); G(k) = 1 - x - x^2/G(k+1), a = 1, b = 1; (continued fraction). - _Sergei N. Gladkovskii_, Dec 08 2011
%F A002426 a(n) = Sum_{k=0..floor(n/3)} (-1)^k*binomial(2*n-3*k-1, n-3*k)*binomial(n, k). - _Gopinath A. R._, Feb 10 2012
%F A002426 G.f.: A(x) = x*B'(x)/B(x) where B(x) satisfies B(x) = x*(1 + B(x) + B(x)^2). - _Vladimir Kruchinin_, Feb 03 2013 (B(x) = x*A001006(x) - _Michael Somos_, Jul 08 2014)
%F A002426 G.f.: G(0), where G(k) = 1 + x*(2 + 3*x)*(4*k + 1)/(4*k + 2 - x*(2 + 3*x)*(4*k + 2)*(4*k + 3)/(x*(2 + 3*x)*(4*k + 3) + 4*(k + 1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 29 2013
%F A002426 E.g.f.: exp(x) * Sum_{k>=0} (x^k/k!)^2. - _Geoffrey Critzer_, Sep 04 2013
%F A002426 G.f.: Sum_{n>=0} (2*n)!/n!^2*(x^(2*n)/(1 - x)^(2*n+1)). - _Paul D. Hanna_, Sep 21 2013
%F A002426 0 = a(n)*(9*a(n+1) + 9*a(n+2) - 6*a(n+3)) + a(n+1)*(3*a(n+1) + 4*a(n+2) - 3*a(n+3)) + a(n+2)*(-a(n+2) + a(n+3)) for all n in Z. - _Michael Somos_, Jul 08 2014
%F A002426 a(n) = hypergeometric([-n/2, (1-n)/2], [1], 4). - _Peter Luschny_, Sep 17 2014
%F A002426 a(n) = A132885(n,0), that is, a(n) = A132885(A002620(n+1)). - _Altug Alkan_, Nov 29 2015
%F A002426 a(n) = GegenbauerC(n,-n,-1/2). - _Peter Luschny_, May 07 2016
%F A002426 a(n) = 4^n*JacobiP[n,-n-1/2,-n-1/2,-1/2]. - _Peter Luschny_, May 13 2016
%F A002426 From _Alexander Burstein_, Oct 03 2017: (Start)
%F A002426 G.f.: A(4*x) = B(-x)*B(3*x), where B(x) is the g.f. of A000984.
%F A002426 G.f.: A(2*x)*A(-2*x) = B(x^2)*B(9*x^2).
%F A002426 G.f.: A(x) = 1 + x*M'(x)/M(x), where M(x) is the g.f. of A001006. (End)
%F A002426 a(n) = Sum_{i=0..n/2} n!/((n - 2*i)!*(i!)^2). [Cf. Lalo and Lalo link. It is Luschny's terminating hypergeometric sum.] - _Shara Lalo_ and _Zagros Lalo_, Oct 03 2018
%F A002426 From _Peter Bala_, Feb 07 2022: (Start)
%F A002426 a(n)^2 = Sum_{k = 0..n} (-3)^(n-k)*binomial(2*k,k)^2*binomial(n+k,n-k) and has g.f. Sum_{n >= 0} binomial(2*n,n)^2*x^n/(1 + 3*x)^(2*n+1). Compare with the g.f. for a(n) given above by Hanna.
%F A002426 The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all prime p and positive integers n and k.
%F A002426 Conjecture: The stronger congruences a(n*p^k) == a(n*p^(k-1)) (mod p^(2*k)) hold for all prime p >= 5 and positive integers n and k. (End)
%F A002426 a(n) = A005043(n) + A005717(n) for n >= 1. - _Amiram Eldar_, May 17 2024
%F A002426 For even n, a(n) = (n-1)!!* 2^{n/2}/ (n/2)!* 2F1(-n/2,-n/2;1/2;1/4). For odd n, a(n) = n!! *2^(n/2-1/2) / (n/2-1/2)! * 2F1(1/2-n/2,1/2-n/2;3/2;1/4). - _R. J. Mathar_, Mar 19 2025
%e A002426 For n = 2, (x^2 + x + 1)^2 = x^4 + 2*x^3 + 3*x^2 + 2*x + 1, so a(2) = 3. - _Michael B. Porter_, Sep 06 2016
%p A002426 A002426 := proc(n) local k;
%p A002426 sum(binomial(n, k)*binomial(n-k, k), k=0..floor(n/2));
%p A002426 end proc: # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001
%p A002426 # Alternatively:
%p A002426 a := n -> simplify(GegenbauerC(n,-n,-1/2)):
%p A002426 seq(a(n), n=0..29); # _Peter Luschny_, May 07 2016
%t A002426 Table[ CoefficientList[ Series[(1 + x + x^2)^n, {x, 0, n}], x][[ -1]], {n, 0, 27}] (* _Robert G. Wilson v_ *)
%t A002426 a=b=1; Join[{a,b}, Table[c=((2n-1)b + 3(n-1)a)/n; a=b; b=c; c, {n,2,100}]]; Table[Sqrt[-3]^n LegendreP[n,1/Sqrt[-3]],{n,0,26}] (* _Wouter Meeussen_, Feb 16 2013 *)
%t A002426 a[ n_] := If[ n < 0, 0, 3^n Hypergeometric2F1[ 1/2, -n, 1, 4/3]]; (* _Michael Somos_, Jul 08 2014 *)
%t A002426 Table[4^n *JacobiP[n,-n-1/2,-n-1/2,-1/2], {n,0,29}] (* _Peter Luschny_, May 13 2016 *)
%t A002426 a[n_] := a[n] = Sum[n!/((n - 2*i)!*(i!)^2), {i, 0, n/2}]; Table[a[n], {n, 0, 29}] (* _Shara Lalo_ and _Zagros Lalo_, Oct 03 2018 *)
%o A002426 (PARI) {a(n) = if( n<0, 0, polcoeff( (1 + x + x^2)^n, n))};
%o A002426 (PARI) /* as lattice paths: same as in A092566 but use */
%o A002426 steps=[[2, 0], [0, 2], [1, 1]];
%o A002426 /* _Joerg Arndt_, Jul 01 2011 */
%o A002426 (PARI) a(n)=polcoeff(sum(m=0, n, (2*m)!/m!^2 * x^(2*m) / (1-x+x*O(x^n))^(2*m+1)), n) \\ _Paul D. Hanna_, Sep 21 2013
%o A002426 (Maxima) trinomial(n,k):=coeff(expand((1+x+x^2)^n),x,k);
%o A002426 makelist(trinomial(n,n),n,0,12); /* _Emanuele Munarini_, Mar 15 2011 */
%o A002426 (Maxima) makelist(ultraspherical(n,-n,-1/2),n,0,12); /* _Emanuele Munarini_, Dec 20 2016 */
%o A002426 (Magma) P<x>:=PolynomialRing(Integers()); [Max(Coefficients((1+x+x^2)^n)): n in [0..26]]; // _Bruno Berselli_, Jul 05 2011
%o A002426 (Haskell)
%o A002426 a002426 n = a027907 n n -- _Reinhard Zumkeller_, Jan 22 2013
%o A002426 (Sage)
%o A002426 A002426 = lambda n: hypergeometric([-n/2, (1-n)/2], [1], 4)
%o A002426 [simplify(A002426(n)) for n in (0..29)]
%o A002426 # _Peter Luschny_, Sep 17 2014
%o A002426 (Sage)
%o A002426 def A():
%o A002426 a, b, n = 1, 1, 1
%o A002426 yield a
%o A002426 while True:
%o A002426 yield b
%o A002426 n += 1
%o A002426 a, b = b, ((3 * (n - 1)) * a + (2 * n - 1) * b) // n
%o A002426 A002426 = A()
%o A002426 print([next(A002426) for _ in range(30)]) # _Peter Luschny_, May 16 2016
%o A002426 (Python)
%o A002426 from math import comb
%o A002426 def A002426(n): return sum(comb(n,k)*comb(k,n-k) for k in range(n+1)) # _Chai Wah Wu_, Nov 15 2022
%Y A002426 INVERT transform is A007971. Partial sums are A097893. Squares are A168597.
%Y A002426 Main column of A027907. Column k=2 of A305161. Column k=0 of A328347. Column 1 of A201552(?).
%Y A002426 Cf. A001006, A002878, A005043, A005717, A082758 (bisection), A273055 (bisection), A102445, A113302, A113303, A113304, A113305 (divisibility of central trinomial coefficients), A152227, A277640.
%K A002426 nonn,nice,core,easy
%O A002426 0,3
%A A002426 _N. J. A. Sloane_, _Simon Plouffe_