cp's OEIS Frontend

A001850 Central Delannoy numbers: a(n) = Sum_{k=0..n} C(n,k)*C(n+k,k).

%I A001850 M2942 N1184 #493 Jul 16 2025 23:02:10
%S A001850 1,3,13,63,321,1683,8989,48639,265729,1462563,8097453,45046719,
%T A001850 251595969,1409933619,7923848253,44642381823,252055236609,
%U A001850 1425834724419,8079317057869,45849429914943,260543813797441,1482376214227923,8443414161166173,48141245001931263
%N A001850 Central Delannoy numbers: a(n) = Sum_{k=0..n} C(n,k)*C(n+k,k).
%C A001850 Number of paths from (0,0) to (n,n) in an n X n grid using only steps north, northeast and east (i.e., steps (1,0), (1,1), and (0,1)).
%C A001850 Also the number of ways of aligning two sequences (e.g., of nucleotides or amino acids) of length n, with at most 2*n gaps (-) inserted, so that while unnecessary gappings: - -a a- - are forbidden, both b- and -b are allowed. (If only other of the latter is allowed, then the sequence A000984 gives the number of alignments.) There is an easy bijection from grid walks given by Dickau to such set of alignments (e.g., the straight diagonal corresponds to the perfect alignment with no gaps). - _Antti Karttunen_, Oct 10 2001
%C A001850 Also main diagonal of array A008288 defined by m(i,1) = m(1,j) = 1, m(i,j) = m(i-1,j-1) + m(i-1,j) + m(i,j-1). - _Benoit Cloitre_, May 03 2002
%C A001850 So, as a special case of _Dmitry Zaitsev_'s Dec 10 2015 comment on A008288, a(n) is the number of points in Z^n that are L1 (Manhattan) distance <= n from any given point. These terms occur in the crystal ball sequences: a(n) here is the n-th term in the sequence for the n-dimensional cubic lattice. See A008288 for a list of crystal ball sequences (rows or columns of A008288). - _Shel Kaphan_, Dec 26 2022
%C A001850 a(n) is the number of n-matchings of a comb-like graph with 2*n teeth. Example: a(2) = 13 because the graph consisting of a horizontal path ABCD and the teeth Aa, Bb, Cc, Dd has 13 2-matchings: any of the six possible pairs of teeth and {Aa, BC}, {Aa, CD}, {Bb, CD}, {Cc, AB}, {Dd, AB}, {Dd, BC}, {AB, CD}. - _Emeric Deutsch_, Jul 02 2002
%C A001850 Number of ordered trees with 2*n+1 edges, having root of odd degree, nonroot nodes of outdegree at most 2 and branches of odd length. - _Emeric Deutsch_, Aug 02 2002
%C A001850 The sum of the first n coefficients of ((1 - x) / (1 - 2*x))^n is a(n-1). - _Michael Somos_, Sep 28 2003
%C A001850 Row sums of A063007 and A105870. - _Paul Barry_, Apr 23 2005
%C A001850 The Hankel transform (see A001906 for definition) of this sequence is A036442: 1, 4, 32, 512, 16384, ... . - _Philippe Deléham_, Jul 03 2005
%C A001850 Also number of paths from (0,0) to (n,0) using only steps U = (1,1), H = (1,0) and D =(1,-1), U can have 2 colors and H can have 3 colors. - _N-E. Fahssi_, Jan 27 2008
%C A001850 Equals row sums of triangle A152250 and INVERT transform of A109980: (1, 2, 8, 36, 172, 852, ...). - _Gary W. Adamson_, Nov 30 2008
%C A001850 Number of overpartitions in the n X n box (treat a walk of the type in the first comment as an overpartition, by interpreting a NE step as N, E with the part thus created being overlined). - _William J. Keith_, May 19 2017
%C A001850 Diagonal of rational functions 1/(1 - x - y - x*y), 1/(1 - x - y*z - x*y*z). - _Gheorghe Coserea_, Jul 03 2018
%C A001850 Dimensions of endomorphism algebras End(R^{(n)}) in the Delannoy category attached to the oligomorphic group of order preserving self-bijections of the real line. - _Noah Snyder_, Mar 22 2023
%C A001850 a(n) is the number of ways to tile a strip of length n with white squares, black squares, and red dominos, where we must have an equal number of white and black squares. - _Greg Dresden_ and Leo Zhang, Jul 11 2025
%D A001850 Frits Beukers, Arithmetic properties of Picard-Fuchs equations, Séminaire de Théorie des nombres de Paris, 1982-83, Birkhäuser Boston, Inc.
%D A001850 Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 593.
%D A001850 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
%D A001850 L. Moser and W. Zayachkowski, Lattice paths with diagonal steps, Scripta Math., 26 (1961), 223-229.
%D A001850 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001850 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A001850 R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 2, 1999; see Example 6.3.8 and Problem 6.49.
%D A001850 D. B. West, Combinatorial Mathematics, Cambridge, 2021, p. 28.
%H A001850 Chai Wah Wu, <a href="/A001850/b001850.txt">Table of n, a(n) for n = 0..1308</a> (all terms < 10^1000, first 201 terms from T. D. Noe)
%H A001850 M. Abrate, S. Barbero, U. Cerruti, and N. Murru, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barbero/barbero9.html">Fixed Sequences for a Generalization of the Binomial Interpolated Operator and for some Other Operators</a>, J. Int. Seq. 14 (2011), #11.8.1.
%H A001850 B. Adamczewski, J. P. Bell, and E. Delaygue, <a href="http://arxiv.org/abs/1603.04187">Algebraic independence of G-functions and congruences "à la Lucas"</a>, arXiv:1603.04187 [math.NT], 2016.
%H A001850 J.-M. Autebert, A.-M. Décaillot, and S. R. Schwer, <a href="https://web.archive.org/web/20160303190600/http://smf4.emath.fr/Publications/Gazette/2003/95/smf_gazette_95_51-62.pdf">H.-A. Delannoy et les oeuvres posthumes d'Édouard Lucas</a>, Gazette des Mathématiciens - no 95, Jan 2003 (in French).
%H A001850 J.-M. Autebert, M. Latapy, and S. R. Schwer, <a href="http://dx.doi.org/10.1016/S0012-365X(02)00351-5">Le treillis des Chemins de Delannoy</a>, Discrete Math., 258 (2002), 225-234.
%H A001850 J.-M. Autebert and S. R. Schwer, <a href="http://dx.doi.org/10.1137/S0895480101387406">On generalized Delannoy paths</a>, SIAM J. Discrete Math., 16(2) (2003), 208-223.
%H A001850 Cyril Banderier and Sylviane Schwer, <a href="https://arxiv.org/abs/math/0411128">Why Delannoy numbers?</a>, arXiv:math/0411128 [math.CO], 2004.
%H A001850 Cyril Banderier and Sylviane Schwer, <a href="https://doi.org/10.1016/j.jspi.2005.02.004">Why Delannoy numbers?</a>, Journal of Statistical Planning and Inference, 135(1) (2005), 40-54.
%H A001850 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Barry/barry91.html">On Integer-Sequence-Based Constructions of Generalized Pascal Triangles</a>, Journal of Integer Sequences, 9 (2006), #06.2.4.
%H A001850 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Barry1/barry242.html">On the Central Coefficients of Riordan Matrices</a>, Journal of Integer Sequences, 16 (2013), #13.5.1.
%H A001850 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Barry3/barry252.html">On the Inverses of a Family of Pascal-Like Matrices Defined by Riordan Arrays</a>, Journal of Integer Sequences, 16 (2013), #13.5.6.
%H A001850 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 A001850 Paul Barry and Aoife Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Barry2/barry190r.html">Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths</a>, Journal of Integer Sequences, 15 (2012), #12.4.8.
%H A001850 Paul Barry, <a href="https://arxiv.org/abs/2307.00098">Moment sequences, transformations, and Spidernet graphs</a>, arXiv:2307.00098 [math.CO], 2023.
%H A001850 Thomas Baruchel and C. Elsner, <a href="http://arxiv.org/abs/1602.06445">On error sums formed by rational approximations with split denominators</a>, arXiv:1602.06445 [math.NT], 2016.
%H A001850 H. Bateman, <a href="/A002692/a002692.pdf">Some problems in potential theory</a>, Messenger Math., 52 (1922), 71-78. [Annotated scanned copy]
%H A001850 Raymond A. Beauregard and Vladimir A. Dobrushkin, <a href="http://www.jstor.org/stable/10.4169/math.mag.89.5.359">Powers of a Class of Generating Functions</a>, Mathematics Magazine, 89(5) (2016), 359-363.
%H A001850 Hacène Belbachir and Abdelghani Mehdaoui, <a href="https://doi.org/10.2989/16073606.2020.1729269">Recurrence relation associated with the sums of square binomial coefficients</a>, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
%H A001850 Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Szalay/szalay42.html">Diagonal Sums in the Pascal Pyramid, II: Applications</a>, J. Int. Seq., 22 (2019), #19.3.5.
%H A001850 A. Bostan, S. Boukraa, J.-M. Maillard, and J.-A. Weil, <a href="http://arxiv.org/abs/1507.03227">Diagonals of rational functions and selected differential Galois groups</a>, arXiv:1507.03227 [math-ph], 2015.
%H A001850 J. S. Caughman et al., <a href="http://dx.doi.org/10.1016/j.disc.2007.05.017">A note on lattice chains and Delannoy numbers</a>, Discrete Math., 308 (2008), 2623-2628.
%H A001850 Jia-Yu Chen and Chen Wang, <a href="https://arxiv.org/abs/2012.04523">Congruences concerning generalized central trinomial coefficients</a>, arXiv:2012.04523 [math.NT], 2020.
%H A001850 Johann Cigler, <a href="http://arxiv.org/abs/1109.1449">Some nice Hankel determinants</a>, arXiv:1109.1449 [math.CO], 2011.
%H A001850 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 A001850 M. Coster, <a href="/A001850/a001850_1.pdf">Email, Nov 1990</a>.
%H A001850 F. D. Cunden, <a href="http://arxiv.org/abs/1412.2172">Statistical distribution of the Wigner-Smith time-delay matrix for chaotic cavities</a>, arXiv:1412.2172 [cond-mat.mes-hall], 2014.
%H A001850 Emeric 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 A001850 Ömür Deveci and Anthony G. Shannon, <a href="https://doi.org/10.20948/mathmontis-2021-50-4">Some aspects of Neyman triangles and Delannoy arrays</a>, Mathematica Montisnigri (2021) Vol. L, 36-43.
%H A001850 R. M. Dickau, <a href="http://mathforum.org/advanced/robertd/delannoy.html">Delannoy and Motzkin Numbers</a> [Many illustrations].
%H A001850 R. M. Dickau, <a href="/A001850/a001850.gif">The 13 paths in a 4 X 4 grid</a>.
%H A001850 T. Doslic, <a href="https://doi.org/10.1007/s10440-009-9515-4">Seven lattice paths to log-convexity</a>, Acta Appl. Mathem. 110(3) (2010), 1373-1392.
%H A001850 Tomislav Došlic and Darko Veljan, <a href="http://dx.doi.org/10.1016/j.disc.2007.04.066">Logarithmic behavior of some combinatorial sequences</a>, Discrete Math. 308(11) (2008), 2182-2212. MR2404544 (2009j:05019) - _N. J. A. Sloane_, May 01 2012
%H A001850 D. Drake, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Drake/drake.html">Bijections from Weighted Dyck Paths to Schröder Paths</a>, J. Int. Seq. 13 (2010), #10.9.2.
%H A001850 Rui Duarte and António Guedes de Oliveira, <a href="https://www.cmup.pt/sites/default/files/2023-08/GF_LP_corrected_0.pdf">Generating functions of lattice paths</a>, Univ. do Porto (Portugal 2023).
%H A001850 M. Dziemianczuk, <a href="http://www.emis.de/journals/INTEGERS/papers/n54/n54.Abstract.html">Generalizing Delannoy numbers via counting weighted lattice paths</a>, INTEGERS, 13 (2013), #A54.
%H A001850 M. Dziemianczuk, <a href="http://arxiv.org/abs/1410.5747">On Directed Lattice Paths With Additional Vertical Steps</a>, arXiv:1410.5747 [math.CO], 2014.
%H A001850 James East and Nicholas Ham, <a href="https://arxiv.org/abs/1811.05735">Lattice paths and submonoids of Z^2</a>, arXiv:1811.05735 [math.CO], 2018.
%H A001850 Steffen Eger, <a href="http://arxiv.org/abs/1511.00622">On the Number of Many-to-Many Alignments of N Sequences</a>, arXiv:1511.00622 [math.CO], 2015.
%H A001850 Luca Ferrari and Emanuele Munarini, <a href="http://arxiv.org/abs/1203.6792">Enumeration of edges in some lattices of paths</a>, arXiv:1203.6792 [math.CO], 2012.
%H A001850 Luca Ferrari and Emanuele Munarini, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Ferrari/ferrari.html">Enumeration of edges in some lattices of paths</a>, J. Int. Seq. 17 (2014), #14.1.5.
%H A001850 Seth Finkelstein, <a href="/A001850/a001850.pdf">Letter to N. J. A. Sloane, Mar 24 1990</a>, with attachments.
%H A001850 S. Garrabrant and I. Pak, <a href="http://arxiv.org/abs/1407.8222">Counting with irrational tiles</a>, arXiv:1407.8222 [math.CO], 2014.
%H A001850 Joël Gay and Vincent Pilaud, <a href="https://arxiv.org/abs/1804.06572">The weak order on Weyl posets</a>, arXiv:1804.06572 [math.CO], 2018.
%H A001850 Taras Goy and Mark Shattuck, <a href="https://doi.org/10.26493/2590-9770.1645.d36">Determinant identities for the Catalan, Motzkin and Schröder numbers</a>, Art Disc. Appl. Math. (2023).
%H A001850 Nate Harman, Andrew Snowden, and Noah Snyder, <a href="https://arxiv.org/abs/2211.15392">The Delannoy Category</a>, arxiv:2211.15392 [math.RT], 2023.
%H A001850 Tian-Xiao He, <a href="https://arxiv.org/abs/2011.00173">One-pth Riordan Arrays in the Construction of Identities</a>, arXiv:2011.00173 [math.CO], 2020.
%H A001850 M. D. Hirschhorn, <a href="http://web.maths.unsw.edu.au/~mikeh/webpapers/paper68.ps">How many ways can a king cross the board?</a>, Austral. Math. Soc. Gaz., 27 (2000), 104-106.
%H A001850 Nick Hobson, <a href="/A001850/a001850.py.txt">Python program for this sequence</a>.
%H A001850 P.-Y. Huang, S.-C. Liu, and Y.-N. 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 A001850 Milan Janjic, <a href="https://old.pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>
%H A001850 Svante Janson, <a href="https://arxiv.org/abs/1804.06071">Patterns in random permutations avoiding some sets of multiple patterns</a>, arXiv:1804.06071 [math.PR], 2018.
%H A001850 S. Kaparthi and H. R. Rao, <a href="http://dx.doi.org/10.1016/0166-218X(91)90055-2">Higher dimensional restricted lattice paths with diagonal steps</a>, Discr. Appl. Math., 31 (1991), 279-289.
%H A001850 D. E. Knuth and N. J. A. Sloane, <a href="/A047662/a047662.pdf">Correspondence, December 1999</a>.
%H A001850 Daniel Krenn and Jeffrey Shallit, <a href="https://arxiv.org/abs/2401.14231">Strongly k-recursive sequences</a>, arXiv:2401.14231 [cs.FL], 2024.
%H A001850 D. F. Lawden, <a href="http://www.jstor.org/stable/3608257">On the Solution of Linear Difference Equations</a>, Math. Gaz., 36 (1952), 193-196.
%H A001850 Tamás Lengyel, <a href="http://math.colgate.edu/~integers/v86/v86.pdf">On some p-adic properties and supercongruences of Delannoy and Schröder Numbers</a>, Integers (2021) Vol. 21, #A86.
%H A001850 Huyile Liang, Yanni Pei, and Yi Wang, <a href="https://arxiv.org/abs/2302.11856">Analytic combinatorics of coordination numbers of cubic lattices</a>, arXiv:2302.11856 [math.CO], 2023. See p. 4.
%H A001850 Huyile Liang, Jeffrey Remmel, and Sainan Zheng, <a href="https://arxiv.org/abs/1710.05795">Stieltjes moment sequences of polynomials</a>, arXiv:1710.05795 [math.CO], 2017, see page 18.
%H A001850 Max A. Little and Ugur Kayas, <a href="https://arxiv.org/abs/2107.01752">Polymorphic dynamic programming by algebraic shortcut fusion</a>, arXiv:2107.01752 [cs.DS], 2021.
%H A001850 Lily L. Liu, <a href="https://doi.org/10.37236/329">Positivity of three-term recurrence sequences</a>, Electronic J. Combinatorics, 17 (2010), #R57.
%H A001850 Romeo Mestrovic, <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 A001850 Lili Mu and Sai-nan Zheng, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Zheng/zheng8.html">On the Total Positivity of Delannoy-Like Triangles</a>, Journal of Integer Sequences, 20 (2017), #17.1.6.
%H A001850 Leo Moser, <a href="http://www.jstor.org/stable/3611095">Note 2487: King paths on a chessboard</a>, Math. Gaz., 39 (1955), 54 (one page only).
%H A001850 Emanuele Munarini, <a href="http://www.emis.de/journals/INTEGERS/papers/j29/j29.Abstract.html">Combinatorial properties of the antichains of a garland</a>, Integers, 9 (2009), 353-374.
%H A001850 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, 9 (2006), #06.2.7.
%H A001850 P. Peart and W.-J. 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., 3 (2000), #00.2.1.
%H A001850 R. Pemantle and M. C. Wilson, <a href="http://arXiv.org/abs/math.CO/0003192">Asymptotics of multivariate sequences, I: smooth points of the singular variety</a>, arXiv:math/0003192 [math.CO], 2000.
%H A001850 C. de Jesús 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 A001850 F. Qi, X.-T. Shi and B.-N. Guo, <a href="https://doi.org/10.1007/s13226-016-0211-6">Some properties of the Schroder numbers</a>, Indian J. Pure Appl. Math 47 (4) (2016) 717-732.
%H A001850 J. L. Ramírez and V. 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 A001850 J. Riordan, <a href="/A001850/a001850_2.pdf">Letter, Jul 06 1978</a>.
%H A001850 John Riordan, <a href="/A002720/a002720_3.pdf">Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences</a>. Note that the sequences are identified by their N-numbers, not their A-numbers.
%H A001850 E. Rowland and D. Zeilberger, <a href="http://arxiv.org/abs/1311.4776">A Case Study in Meta-AUTOMATION: AUTOMATIC Generation of Congruence AUTOMATA For Combinatorial Sequences</a>, arXiv:1311.4776 [math.CO], 2013.
%H A001850 S. Samieinia, <a href="http://www.math.su.se/reports/2008/3">The number of continuous curves in digital geometry</a>, Research Reports in Mathematics, Number 3, 2008.
%H A001850 S. R. Schwer and J.-M. Auterbert, <a href="https://www.researchgate.net/publication/30449392_Henri-Auguste_Delannoy_une_biographie">Henri-Auguste Delannoy, une biographie</a>, Math. & Sci. hum. / Mathematical Social Sciences, 43e année, no. 174 (2006), 25-67.
%H A001850 Seunghyun Seo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Seo/seo2.html">The Catalan Threshold Arrangement</a>, Journal of Integer Sequences, 20 (2017), #17.1.1.
%H A001850 T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/series006">The Delannoy numbers</a>.
%H A001850 Mark Shattuck, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_42_from93to101.pdf">On the zeros of some polynomials with combinatorial coefficients</a>, Annales Mathematicae et Informaticae, 42 (2013), 93-101.
%H A001850 Zhao Shen, <a href="https://arxiv.org/abs/2112.11135">On the 3-adic valuation of a class of Apery-like numbers</a>, arXiv:2112.11135 [math.NT], 2021.
%H A001850 J. B. Slowinski, <a href="http://dx.doi.org/10.1006/mpev.1998.0522">The Number of Multiple Alignments</a>, Molecular Phylogenetics and Evolution, 10(2) (1998), 264-266.
%H A001850 J. B. Slowinski, <a href="http://www.neurociencias.org.ve/cont-cursos-laboratorio-de-neurociencias-luz/Slowinski1998%20phylogenetics.pdf">The Number of Multiple Alignments</a>, Molecular Phylogenetics and Evolution, 10(2) (1998), 264-266.
%H A001850 R. G. Stanton and D. D. Cowan, <a href="http://dx.doi.org/10.1137/1012049">Note on a "square" functional equation</a>, SIAM Rev., 12 (1970), 277-279.
%H A001850 R. A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/SULANKE/sulanke.html">Moments of generalized Motzkin paths</a>, J. Integer Sequences, 3 (2000), #00.1.
%H A001850 R. A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Sulanke/delannoy.html">Objects Counted by the Central Delannoy Numbers</a>, Journal of Integer Sequences, 5 (2002), #03.1.5.
%H A001850 R. A. Sulanke et al., <a href="http://www.jstor.org/stable/3647842">Another description of the central Delannoy numbers, Problem 10894</a>, Amer. Math. Monthly, 110(5) (2003), 443-444.
%H A001850 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1009.2486">On Delannoy numbers and Schroeder numbers</a>, Journal of Number Theory 131(12) (2011), 2387-2397. doi:<a href="http://dx.doi.org/10.1016/j.jnt.2011.06.005">10.1016/j.jnt.2011.06.005</a>
%H A001850 Marius Tarnauceanu, <a href="http://dx.doi.org/10.1016/j.ejc.2007.12.005">The number of fuzzy subgroups of finite cyclic groups and Delannoy numbers</a>, European Journal of Combinatorics 30(1) (2009), 283-287.
%H A001850 Bernhard Von Stengel, <a href="https://doi.org/10.1007/PL00009438">New maximal numbers of equilibria in bimatrix games</a>, Discrete & Computational Geometry 21(4) (1999), 557-568. See Eq. (3.7).
%H A001850 Rong-Hua Wang and Michael X. X. Zhong, <a href="https://arxiv.org/abs/2505.05728">Congruences for sums of Delannoy numbers and polynomials</a>, arXiv:2505.05728 [math.CO], 2025. See p. 2.
%H A001850 Yi Wang, Sai-Nan Zheng, and Xi Chen, <a href="https://doi.org/10.1016/j.disc.2019.04.003">Analytic aspects of Delannoy numbers</a>, Discrete Mathematics 342.8 (2019): 2270-2277.
%H A001850 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DelannoyNumber.html">Delannoy Number</a>.
%H A001850 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SchmidtsProblem.html">Schmidt's Problem</a>.
%H A001850 W.-J. Woan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WOAN/hankel2.html">Hankel Matrices and Lattice Paths</a>, J. Integer Sequences, 4 (2001), #01.1.2.
%H A001850 E. X. W. Xia and O. X. M. Yao, <a href="https://doi.org/10.37236/3412">A Criterion for the Log-Convexity of Combinatorial Sequences</a>, The Electronic Journal of Combinatorics, 20 (2013), #P3.
%H A001850 Lin Yang, Yu-Yuan Zhang, and Sheng-Liang Yang, <a href="https://doi.org/10.1016/j.laa.2023.12.021">The halves of Delannoy matrix and Chung-Feller properties of the m-Schröder paths</a>, Linear Alg. Appl. (2024).
%F A001850 a(n) = P_n(3), where P_n is n-th Legendre polynomial.
%F A001850 G.f.: 1 / sqrt(1 - 6*x + x^2).
%F A001850 a(n) = a(n-1) + 2*A002002(n) = Sum_{j} A063007(n, j). - _Henry Bottomley_, Jul 02 2001
%F A001850 Dominant term in asymptotic expansion is binomial(2*n, n)/2^(1/4)*((sqrt(2) + 1)/2)^(2*n + 1)*(1 + c_1/n + c_2/n^2 + ...). - _Michael David Hirschhorn_
%F A001850 a(n) = Sum_{i=0..n} (A000079(i)*A008459(n, i)) = Sum_{i=0..n} (2^i * C(n, i)^2). - _Antti Karttunen_, Oct 10 2001
%F A001850 a(n) = Sum_{k=0..n} C(n+k, n-k)*C(2*k, k). - _Benoit Cloitre_, Feb 13 2003
%F A001850 a(n) = Sum_{k=0..n} C(n, k)^2 * 2^k. - _Michael Somos_, Oct 08 2003
%F A001850 a(n - 1) = coefficient of x^n in A120588(x)^n if n>=0. - _Michael Somos_, Apr 11 2012
%F A001850 G.f. of a(n-1) = 1 / (1 - x / (1 - 2*x / (1 - 2*x / (1 - x / (1 - 2*x / (1 - x / ...)))))). - _Michael Somos_, May 11 2012
%F A001850 INVERT transform is A109980. BINOMIAL transform is A080609. BINOMIAL transform of A006139. PSUM transform is A089165. PSUMSIGN transform is A026933. First backward difference is A110170. - _Michael Somos_, May 11 2012
%F A001850 E.g.f.: exp(3*x)*BesselI(0, 2*sqrt(2)*x). - _Vladeta Jovovic_, Mar 21 2004
%F A001850 a(n) = Sum_{k=0..n} C(2*n-k, n)*C(n, k). - _Paul Barry_, Apr 23 2005
%F A001850 a(n) = Sum_{k>=n} binomial(k, n)^2/2^(k+1). - _Vladeta Jovovic_, Aug 25 2006
%F A001850 a(n) = a(-1 - n) for all n in Z. - _Michael Somos_, Sep 23 2006
%F A001850 D-finite with recurrence: a(-1) = a(0) = 1; n*a(n) = 3*(2*n-1)*a(n-1) - (n-1)*a(n-2). Eq (4) in _T. D. Noe_'s article in JIS 9 (2006) #06.2.7.
%F A001850 Define general Delannoy numbers by (i,j > 0): d(i,0) = d(0,j) = 1 =: d(0,0) and d(i,j) = d(i-1,j-1) + d(i-2,j-1) + d(i-1,j). Then a(k) = Sum_{j >= 0} d(k,j)^2 + d(k-1,j)^2 = A026933(n)+A026933(n-1). This is a special case of the following formula for general Delannoy numbers: d(k,j) = Sum_{i >= 0, p=0..n} d(p, i) * d(n-p, j-i) + d(p-1, i) * d(n-p-1, j-i-1). - _Peter E John_, Oct 19 2006
%F A001850 Coefficient of x^n in (1 + 3*x + 2*x^2)^n. - _N-E. Fahssi_, Jan 11 2008
%F A001850 a(n) = A008288(A046092(n)). - _Philippe Deléham_, Apr 08 2009
%F A001850 G.f.: 1/(1 - x - 2*x/(1 - x - x/(1 - x - x/(1 - x - x/(1 - ... (continued fraction). - _Paul Barry_, May 28 2009
%F A001850 G.f.: d/dx log(1/(1 - x*A001003(x))). - _Vladimir Kruchinin_, Apr 19 2011
%F A001850 G.f.: 1/(2*Q(0) + x - 1) where Q(k) = 1 + k*(1-x) - x - x*(k + 1)*(k + 2)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Mar 14 2013
%F A001850 a(n) = Sum_{k=0..n} C(n,k) * C(n+k,k). - _Joerg Arndt_, May 11 2013
%F A001850 G.f.: G(0), where G(k) = 1 + x*(6 - x)*(4*k + 1)/(4*k + 2 - 2*x*(6-x)*(2*k + 1)*(4*k + 3)/(x*(6 - x)*(4*k + 3) + 4*(k + 1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 22 2013
%F A001850 G.f.: 2/G(0), where G(k) = 1 + 1/(1 - x*(6 - x)*(2*k - 1)/(x*(6 - x)*(2*k - 1) + 2*(k + 1)/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jul 16 2013
%F A001850 G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(6 - x)*(2*k + 1)/(x*(6 - x)*(2*k + 1) + 2*(k + 1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jul 17 2013
%F A001850 a(n)^2 = Sum_{k=0..n} 2^k * C(2*k, k)^2 * C(n+k, n-k) = A243949(n). - _Paul D. Hanna_, Aug 17 2014
%F A001850 a(n) = hypergeom([-n, -n], [1], 2). - _Peter Luschny_, Nov 19 2014
%F A001850 a(n) = Sum_{k=0..n/2} C(n-k,k) * 3^(n-2*k) * 2^k * C(n,k). - _Vladimir Kruchinin_, Jun 29 2015
%F A001850 a(n) = A049600(n, n-1).
%F A001850 a(n) = Sum_{0 <= j, k <= n} (-1)^(n+j)*C(n,k)*C(n,j)*C(n+k,k)*C(n+k+j,k+j). Cf. A126086 and A274668. - _Peter Bala_, Jan 15 2020
%F A001850 a(n) ~ c * (3 + 2*sqrt(2))^n / sqrt(n), where c = 1/sqrt(4*Pi*(3*sqrt(2)-4)) = 0.572681... (Banderier and Schwer, 2005). - _Amiram Eldar_, Jun 07 2020
%F A001850 a(n+1) = 3*a(n) + 2*Sum_{l=1..n} A006318(l)*a(n-l). [Eq. (1.16) in Qi-Shi-Guo (2016)]
%F A001850 a(n) ~ (1 + sqrt(2))^(2*n+1) / (2^(5/4) * sqrt(Pi*n)). - _Vaclav Kotesovec_, Jan 09 2023
%F A001850 a(n-1) + a(n) = A241023(n) for n >= 1. - _Peter Bala_, Sep 18 2024
%F A001850 a(n) = Sum_{k=0..n} C(n+k, 2*k) * C(2*k, k). - _Greg Dresden_ and Leo Zhang, Jul 11 2025
%e A001850 G.f. = 1 + 3*x + 13*x^2 + 63*x^3 + 321*x^4 + 1683*x^5 + 8989*x^6 + ...
%p A001850 seq(add(multinomial(n+k,n-k,k,k),k=0..n),n=0..20); # _Zerinvary Lajos_, Oct 18 2006
%p A001850 seq(orthopoly[P](n,3), n=0..100); # _Robert Israel_, Nov 03 2015
%t A001850 f[n_] := Sum[ Binomial[n, k] Binomial[n + k, k], {k, 0, n}]; Array[f, 21, 0] (* Or *)
%t A001850 a[0] = 1; a[1] = 3; a[n_] := a[n] = (3(2 n - 1)a[n - 1] - (n - 1)a[n - 2])/n; Array[a, 21, 0] (* Or *)
%t A001850 CoefficientList[ Series[1/Sqrt[1 - 6x + x^2], {x, 0, 20}], x] (* _Robert G. Wilson v_ *)
%t A001850 Table[LegendreP[n, 3], {n, 0, 22}] (* _Jean-François Alcover_, Jul 16 2012, from first formula *)
%t A001850 a[n_] := Hypergeometric2F1[-n, n+1, 1, -1]; Table[a[n], {n, 0, 22}] (* _Jean-François Alcover_, Feb 26 2013 *)
%t A001850 a[ n_] := With[ {m = If[n < 0, -1 - n, n]}, SeriesCoefficient[ (1 - 6 x + x^2)^(-1/2), {x, 0, m}]]; (* _Michael Somos_, Jun 10 2015 *)
%o A001850 (PARI) {a(n) = if( n<0, n = -1 - n); polcoeff( 1 / sqrt(1 - 6*x + x^2 + x * O(x^n)), n)}; /* _Michael Somos_, Sep 23 2006 */
%o A001850 (PARI) {a(n) = if( n<0, n = -1 - n); subst( pollegendre(n), x, 3)}; /* _Michael Somos_, Sep 23 2006 */
%o A001850 (PARI) {a(n) = if( n<0, n = -1 - n); n++; subst( Pol(((1 - x) / (1 - 2*x) + O(x^n))^n), x, 1);} /* _Michael Somos_, Sep 23 2006 */
%o A001850 (PARI) a(n)=if(n<0, 0, polcoeff((1+3*x+2*x^2)^n, n)) \\ _Paul Barry_, Aug 22 2007
%o A001850 (PARI) /* same as in A092566 but use */
%o A001850 steps=[[1,0], [0,1], [1,1]]; /* _Joerg Arndt_, Jun 30 2011 */
%o A001850 (PARI) a(n)=sum(k=0,n,binomial(n,k)*binomial(n+k,k)); \\ _Joerg Arndt_, May 11 2013
%o A001850 (PARI) my(x='x+O('x^30)); Vec(1/sqrt(1 - 6*x + x^2)) \\ _Altug Alkan_, Oct 17 2015
%o A001850 (Python) # from Nick Hobson.
%o A001850 def f(a, b):
%o A001850     if a == 0 or b == 0:
%o A001850         return 1
%o A001850     return f(a, b - 1) + f(a - 1, b) + f(a - 1, b - 1)
%o A001850 [f(n, n) for n in range(7)]
%o A001850 (Python)
%o A001850 from gmpy2 import divexact
%o A001850 A001850 = [1, 3]
%o A001850 for n in range(2,10**3):
%o A001850     A001850.append(divexact(A001850[-1]*(6*n-3)-(n-1)*A001850[-2],n))
%o A001850 # _Chai Wah Wu_, Sep 01 2014
%o A001850 (Maxima) a(n):=coeff(expand((1+3*x+2*x^2)^n),x,n);
%o A001850 makelist(a(n),n,0,12); /* _Emanuele Munarini_, Mar 02 2011 */
%o A001850 (Sage)
%o A001850 a = lambda n: hypergeometric([-n, -n], [1], 2)
%o A001850 [simplify(a(n)) for n in range(23)] # _Peter Luschny_, Nov 19 2014
%Y A001850 Cf. A008288, bisection of A026003, A027618, A047665, A052141, A084773, A152250, A109980, A000129, A078057, A241023, A243949.
%Y A001850 Main diagonal of A064861.
%Y A001850 Column k=2 of A262809 and A263159.
%K A001850 nonn,easy,nice
%O A001850 0,2
%A A001850 _N. J. A. Sloane_
%E A001850 New name and reference Sep 15 1995
%E A001850 Formula and more references from _Don Knuth_, May 15 1996