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 A008288 Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals. 101

%I

%S 1,1,1,1,3,1,1,5,5,1,1,7,13,7,1,1,9,25,25,9,1,1,11,41,63,41,11,1,1,13,

%T 61,129,129,61,13,1,1,15,85,231,321,231,85,15,1,1,17,113,377,681,681,

%U 377,113,17,1,1,19,145,575,1289,1683,1289,575,145,19,1,1,21,181,833,2241,3653,3653

%N Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.

%C Also called the tribonacci triangle [Alladi and Hoggatt]. - _N. J. A. Sloane_, Mar 23 2014

%C D(n-k,k) is the number of lattice paths from (0,0) to (n,k) using steps (1,0), (0,1), (1,1). - _Joerg Arndt_, Jul 01 2011

%C Or, triangle read by rows of coefficients of polynomials P[n](x) defined by P[0] = 1, P[1] = x+1; for n >= 2, P[n] = (x+1)*P[n-1] + x*P[n-2].

%C D(n, k) is the number of k-matchings of a comb-like graph with n+k teeth. Example: D(1, 3)=7 because the graph consisting of a horizontal path ABCD and the teeth Aa, Bb, Cc, Dd has seven 3-matchings: four triples of three teeth and the three triples {Aa, Bb, CD}, {Aa, Dd, BC}, {Cc, Dd, AB}. Also D(3, 1)=7, the 1-matchings of the same graph being the seven edges: {AB}, {BC}, {CD}, {Aa}, {Bb}, {Cc}, {Dd}. - _Emeric Deutsch_, Jul 01 2002

%C Sum of n-th row = A000129(n). - _Reinhard Zumkeller_, Dec 03 2004

%C The A-sequence for this Riordan type triangle (see the P. Barry comment under Formula) is A112478 and the Z-sequence the trivial: {1,0,0,0...}. See the W. Lang link under A006232 for Sheffer a- and z-sequences where also Riordan A- and Z-sequences are explained. This leads to the recurrence for the triangle given below. - _Wolfdieter Lang_, Jan 21 2008

%C Row sums are A000129. - _Roger L. Bagula_, Dec 09 2008

%C The triangle or chess sums, see A180662 for their definitions, link the Delannoy numbers with twelve different sequences, see the crossrefs. All sums come in pairs due to the symmetrical nature of this triangle. The knight sums Kn14 and Kn15 have been added. It is remarkable that all knight sums are related to the tribonacci numbers, that is, A000073 and A001590, but none of the others. - _Johannes W. Meijer_, Sep 22 2010

%C A008288 is jointly generated with A035607 as an array of coefficients of polynomials u(n,x): initially, u(1,x) = v(1,x) = 1; for n > 1, u(n,x) = x*u(n-1,x) + v(n-1) and v(n,x) = 2*x*u(n-1,x) + v(n-1,x). See the Mathematica section. - _Clark Kimberling_, Mar 09 2012

%C Row n, for n>0, of the Roger L. Bagula's triangle in the Example section shows the coefficients of the polynomial u(n) = c(0) + c(1)*x + ... + c(n)*x^n which is the numerator of the n-th convergent of the continued fraction [k, k, k, ... ], where k = sqrt(x) + 1/sqrt(x); see A230000. - _Clark Kimberling_, Nov 13 2013

%C In an n-dimensional hypercube lattice, D(n,k) gives the number of nodes situated at a Minkowski (Manhattan) distance of k from a given node. In cellular automata theory, the cells at Manhattan distance k what is called the is von Neumann neighborhood of radius k. For k=1, see A005843. - _Dmitry Zaitsev_, Dec 10 2015

%D Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 593.

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.

%D Steven Edwards and W. Griffiths, Generalizations of Delannoy and cross polytope numbers, Fib. Q., 55 (2017), 356-366.

%D L. Moser and W. Zayachkowski, Lattice paths with diagonal steps, Scripta Mathematica, 26 (1963) 223-229.

%D Shiva Samieinia, Digital straight line segments and curves. Licentiate Thesis. Stockholm University, Department of Mathematics, Report 2007:6.

%D Seunghyun Seo, The Catalan Threshold Arrangement, Journal of Integer Sequences, 2017 Vol. 20, #17.1.1.

%H T. D. Noe, <a href="/A008288/b008288.txt">Table of n, a(n) for n=0..5150</a>

%H K. Alladi and V. E. Hoggatt Jr., <a href="http://www.fq.math.ca/Scanned/15-1/alladi.pdf">On tribonacci numbers and related functions</a>, Fibonacci Quart. 15 (1977), 42-45.

%H J.-M. Autebert et al., <a href="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 Bela Bajnok, <a href="https://arxiv.org/abs/1705.07444">Additive Combinatorics: A Menu of Research Problems</a>, arXiv:1705.07444 [math.NT], May 2017. See Sect. 2.3.

%H C. Banderier and S. Schwer, <a href="https://arxiv.org/abs/math/0411128">Why Delannoy numbers?</a>, arXiv:math/0411128 [math.CO], 2004.

%H 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, Vol. 9 (2006), Article 06.2.4.

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Barry4/bern2.html">Riordan-Bernstein Polynomials, Hankel Transforms and Somos Sequences</a>, Journal of Integer Sequences, Vol. 15 2012, #12.8.2

%H 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 Paul Barry, <a href="https://arxiv.org/abs/1804.05027">The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays</a>, arXiv:1804.05027 [math.CO], 2018.

%H Frédéric Bihan, Francisco Santos, Pierre-Jean Spaenlehauer, <a href="https://arxiv.org/abs/1804.5683">A Polyhedral Method for Sparse Systems with many Positive Solutions</a>, arXiv:1804.05683 [math.CO], 2018.

%H B. A. Bondarenko, <a href="http://www.fq.math.ca/pascal.html">Generalized Pascal Triangles and Pyramids</a> (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8. English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 37.

%H D. Bump, K. Choi, P. Kurlberg and J. Vaaler, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.23.7311">A local Riemann hypothesis, I</a>, Math. Zeit. 233, (2000), 1-19.

%H C. Carré, N. Debroux, M. Deneufchatel, J.-P. Dubernard et al., <a href="http://hal.archives-ouvertes.fr/docs/00/90/58/89/PDF/polycubes.pdf">Dirichlet convolution and enumeration of pyramid polycubes</a>, 2013.

%H C. Carre, N. Debroux, M. Deneufchatel, J.-Ph. Dubernard, C. Hillariet, J.-G. Luque, O. Mallet, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Dubernard/dub4.html">Enumeration of Polycubes and Dirichlet Convolutions</a>, J. Int. Seq. 18 (2015) 15.11.4

%H 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 H. Delannoy, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k201183v/f73.image">Emploi de l'échiquier pour la résolution de certains problèmes de probabilités</a>, Association Française pour l'Avancement des Sciences, 24th session, 1895. pp. 70-90 (table given on pp. 76).

%H J. R. Dias, <a href="http://www.stkpula.hr/ccacaa/CCA-PDF/cca2004/v77-n1_n2/CCA_77_2004_325-330_dias.pdf">Properties and relationships of conjugated polyenes having a reciprocal eigenvalue spectrum - dendralene and radialene hydrocarbons</a>, Croatica Chem. Acta, 77 (2004), 325-330.

%H R. Feria-Puron, H. Perez-Roses, J. Ryan, <a href="http://arxiv.org/abs/1503.07357">Searching for Large Circulant Graphs</a>, arXiv preprint arXiv:1503.07357 [math.CO], 2015.

%H R. Feria-Purón, J. Ryan, H. Pérez-Rosés, <a href="http://dx.doi.org/10.1016/j.endm.2014.08.031">Searching for Large Multi-Loop Networks</a>, Electronic Notes in Discrete Mathematics, Volume 46, September 2014, pages 233-240.

%H Rebecca Hartman-Baker, <a href="https://www.ideals.illinois.edu/handle/2142/11055">The Diffusion Equation Method for Global Optimization and Its Application to Magnetotelluric Geoprospecting</a>, University of Illinois, Urbana-Champaign, 2005.

%H G. Hetyei, <a href="http://arxiv.org/abs/0909.5512">Shifted Jacobi polynomials and Delannoy numbers</a>, arXiv:0909.5512 [math.CO], 2009. - _Peter Bala_, Oct 28 2008

%H G. Hetyei, <a href="http://www.math.cornell.edu/event/conf/billera65/notes/hetyei.pdf">Links we almost missed between Delannoy numbers and Legendre polynomials</a>. - _Peter Bala_, Nov 10 2008

%H V. E. Hoggatt, Jr., <a href="/A001628/a001628.pdf">Letters to N. J. A. Sloane, 1974-1975</a>

%H M. Janjic and B. Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From N. J. A. Sloane, Feb 13 2013

%H M. Janjic, B. Petkovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Janjic/janjic45.html">A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers</a>, J. Int. Seq. 17 (2014) # 14.3.5

%H 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 G. Kreweras, <a href="http://www.numdam.org/item?id=BURO_1973__20__3_0">Sur les hiérarchies de segments</a>, Cahiers Bureau Universitaire Recherche Opérationnelle, # 20, Inst. Statistiques, Univ. Paris, 1973, pp. 4-10..

%H G. Kreweras, <a href="/A001844/a001844.pdf">Sur les hiérarchies de segments</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)

%H G. Kreweras, <a href="/A006318/a006318_2.pdf">Aires des chemins surdiagonaux et application à un problème économique</a>, Cahiers du Bureau universitaire de recherche opérationnelle Série Recherche 24 (1976): 1-8. [Annotated scanned copy]

%H M. LLadser, <a href="https://arxiv.org/abs/math/0604152">Uniform formulas for coefficients of meromorphic functions</a>, arXiv:math/0604152 [math.CO], 2006.

%H E. Lucas, <a href="http://denise.vella.chemla.free.fr/lucas.pdf">Théorie des Nombres</a>. Gauthier-Villars, Paris, 1891, Vol. 1, p. 174.

%H J. W. Meijer, <a href="https://www.ucbcba.edu.bo/Publicaciones/revistas/actanova/documentos/v4n4/Ensayos_Meijer2010_PI_.3r.pdf">Famous numbers on a chessboard</a>, Acta Nova, Volume 4, No.4, December 2010. pp. 589-598.

%H Mirka Miller, Hebert Perez-Roses, and Joe Ryan, <a href="http://arxiv.org/abs/1203.4069">The Maximum Degree-and-Diameter-Bounded Subgraph in the Mesh</a>, arXiv preprint arXiv:1203.4069 [math.CO], 2012.

%H Lili Mu and Sai-nan Zheng, <a href="http://www.emis.ams.org/journals/JIS/VOL20/Zheng/zheng8.html"> On the Total Positivity of Delannoy-Like Triangles</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.1.6.

%H Richard L. Ollerton and Anthony G. Shannon, <a href="http://www.fq.math.ca/Scanned/36-2/ollerton.pdf">Some properties of generalized Pascal squares and triangles</a>, Fib. Q., 36 (1998), 98-109. See Table 9.

%H L. Pachter and B. Sturmfels, <a href="http://arXiv.org/abs/math.ST/0409132">The mathematics of phylogenomics</a>, arXiv:math/0409132 [math.ST], 2004-2005.

%H R. Pemantle and M. C. Wilson, <a href="https://arxiv.org/abs/math/0003192">Asymptotics of multivariate sequences, I: smooth points of the singular variety</a>, arXiv:math/0003192 [math.CO], 2000.

%H J. L. Ramirez and V. F. Sirvent, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Ramirez/ramirez4.html">Incomplete Tribonacci Numbers and Polynomials</a>, Journal of Integer Sequences, Vol. 17, 2014, #14.4.2. See Table 1. - _N. J. A. Sloane_, Mar 23 2014

%H Marko Razpet, <a href="http://dx.doi.org/10.1016/S0012-365X(01)00098-X">A self-similarity structure generated by king's walk</a>, Algebraic and topological methods in graph theory (Lake Bled, 1999). Discrete Math. 244 (2002), no. 1-3, 423--433. MR1844050 (2002k:05022).

%H S. Samieinia, <a href="http://dx.doi.org/10.4171/PM/1858">The number of continuous curves in digital geometry</a>, Port. Math. 67 (1) (2010) 75-89.

%H Seunghyun Seo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Seo/seo2.html">The Catalan Threshold Arrangement</a>, Journal of Integer Sequences, 2017 Vol. 20, #17.1.1.

%H M. Shattuck, <a href="http://arxiv.org/abs/1406.2755">Combinatorial identities for incomplete tribonacci polynomials</a>, arXiv preprint arXiv:1406.2755 [math.CO], 2014.

%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DelannoyNumber.html">Delannoy Number</a>

%H Dmitry Zaitsev, <a href="https://arxiv.org/abs/1605.08870">k-neighborhood for Cellular Automata</a>, arXiv preprint arXiv:1605.08870 [cs.DM], 2016.

%H Liang Zhao and Fengyao Yan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Zhao/zhao17.html">Note on Total Positivity for a Class of Recursive Matrices</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.6.5.

%F D(n, 0) = 1 = D(0, n) for n >= 0; D(n, k) = D(n, k-1) + D(n-1, k-1) + D(n-1, k).

%F Sum_{n >= 0, k >= 0} D(n, k)*x^n*y^k = 1/(1-x-y-x*y).

%F D(n, k) = Sum_{d} binomial(k, d)*binomial(n+k-d, k) = Sum_{d} 2^d*binomial(n, d)*binomial(k, d).

%F Seen as a triangle read by rows: T(n, 0)=T(n, n)=1 for n>=0 and T(n, k)=T(n-1, k-1)+T(n-2, k-1)+T(n-1, k), 0<k<n and n>1. - _Reinhard Zumkeller_, Dec 03 2004

%F Read as a number triangle, this is the Riordan array (1/(1-x), x(1+x)/(1-x)) with T(n, k)=sum{j=0..n-k, C(n-k, j)C(k, j)2^j}. - _Paul Barry_, Jul 18 2005

%F T(n,k) = sum{j=0..n-k, C(k,j)C(n-j,k)}. - _Paul Barry_, May 21 2006

%F Let y^k(n) be the number of Khalimsky-continuous functions f from [0,n-1] to Z such that f(0)=0 and f(n-1)=k. Then y^k(n)=D(i,j) for i=1/2(n-1-k) and j=1/2(n-1+k) where n-1+k belongs to 2Z. - Shiva Samieinia (shiva(AT)math.su.se), Oct 08 2007

%F Recurrence for triangle from A-sequence (see the W. Lang comment above): T(n,k) = sum(A112478(j)*T(n-1,k-1+j),j=0..n-k), n>=1, k>=1.

%F From _Peter Bala_, Jul 17 2008: (Start)

%F The n-th row of the square array is the crystal ball sequence for the product lattice A_1 x...x A_1 (n copies). A035607 is the table of the associated coordination sequences for these lattices.

%F The polynomial p_n(x) := sum {k = 0..n} 2^k*C(n,k)*C(x,k) = sum {k = 0..n} C(n,k)*C(x+k,n), whose values [p_n(0),p_n(1),p_n(2),... ] give the n-th row of the square array, is the Ehrhart polynomial of the n-dimensional cross polytope (the hyperoctahedron) [BUMP et al., Theorem 6].

%F The first few values are p_0(x) = 1, p_1(x) = 2*x+1, p_2(x) = 2*x^2+2*x+1 and p_3(x) = (4*x^3+6*x^2+8*x+3)/3.

%F The reciprocity law p_n(m) = p_m(n) reflects the symmetry of the table.

%F The polynomial p_n(x) is the unique polynomial solution of the difference equation (x+1)*f(x+1) - x*f(x-1) = (2*n+1)*f(x), normalized so that f(0) = 1.

%F These polynomials have their zeros on the vertical line Re x = -1/2 in the complex plane; that is, the polynomials p_n(x-1), n = 1,2,3,..., satisfy a Riemann hypothesis [BUMP et al., Theorem 4]. The o.g.f. for the p_n(x) is (1+t)^x/(1-t)^(x+1) = 1 + (2*x+1)*t + (2*x^2+2*x+1)*t^2 + ... .

%F The square array of Delannoy numbers has a close connection with the constant log(2). The entries in the n-th row of the array occur in the series acceleration formula log(2) = (1-1/2+1/3-...+(-1)^(n+1)/n) + (-1)^n * sum {k = 1..inf} (-1)^(k+1)/(k*T(n,k-1)*T(n,k)).

%F For example, the fourth row of the table (n = 3) gives the series log(2) = 1 - 1/2 + 1/3 - 1/(1*1*7) + 1/(2*7*25) - 1/(3*25*63) + 1/(4*63*129) - ... . See A142979 for further details.

%F Also the main diagonal entries (the central Delannoy numbers) give the series acceleration formula sum {n = 1..inf} 1/(n*T(n-1,n-1)*T(n,n)) = 1/2*log(2), a result due to Burnside.

%F Similar relations hold between log(2) and the crystal ball sequences of the C_n lattices A142992. For corresponding results for the constants zeta(2) and zeta(3), involving the crystal ball sequences for root lattices of type A_n and A_n x A_n, see A108625 and A143007 respectively. (End)

%F From _Peter Bala_, Oct 28 2008: (Start)

%F Hilbert transform of Pascal's triangle A007318 (see A145905 for the definition of this term).

%F T(n+a,n) = P_n(a,0;3) for all integer a such that a >= -n, where P_n(a,0;x) is the Jacobi polynomial with parameters (a,0) [Hetyei]. The related formula A(n,k) = P_k(0,n-k;3) defines the table of asymmetric Delannoy numbers, essentially A049600. (End)

%F Seen as a triangle read by rows: T(n,k) = (-1)^(n-k)*Hyper2F1([-n+k, k+1], [1], 2) for 0<=k<=n. - _Peter Luschny_, Aug 02 2014

%F From _Peter Bala_, Jun 25 2015: (Start)

%F O.g.f. for triangle: A(z,t) = 1/(1 - (1 + t)*z - t*z^2) = 1 + (1 + t)*z + (1 + 3*t + t^2)*z^2 + (1 + 5*t + 5*t^2 + t^3)*z^3 + ....

%F 1 + z*d/dz(A(z,t))/A(z,t) is the o.g.f. for A102413. (End)

%F E.g.f. for the n-th subdiagonal, n = 0,1,2,..., equals exp(x)*P(n,x), where P(n,x) is the polynomial Sum_{k = 0..n} binomial(n,k)*(2*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 + 4*x + 4*x^2/2) = 1 + 5*x + 13*x^2/2! + 25*x^3/3! + 41*x^4/4! + 61*x^5/5! + .... - _Peter Bala_, Mar 05 2017

%F Let a_i(n) be multiplicative with a_i(p^e) = D(i, e), p prime and e >= 0, then Sum_{n>0} a_i(n)/n^s = (zeta(s))^(2*i+1)/(zeta(2*s))^i for i >= 0. - _Werner Schulte_, Feb 14 2018

%F Seen as a triangle read by rows: T(n,k) = Sum_{i=0..k} binomial(n-i,i) * binomial(n-2*i,k-i) for 0<=k<=n. - _Werner Schulte_, Jan 09 2019

%e Square array D(i,j) begins:

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... = A000012

%e 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, ... = A005408

%e 1, 5, 13, 25, 41, 61, 85, 113, 145, 181, ... = A001844

%e 1, 7, 25, 63, 129, 231, 377, 575, 833, 1159, ... = A001845

%e 1, 9, 41, 129, 321, 681, 1289, 2241, 3649, 5641, ... = A001846

%e ...

%e As a triangular array (on its side) this begins

%e 0, 0, 0, 0, 0, 1, 1, 11, 11, ...

%e 0, 0, 0, 0, 1, 1, 9, 9, 61, ...

%e 0, 0, 0, 1, 1, 7, 7, 41, 41, ...

%e 0, 0, 1, 1, 5, 5, 25, 25, 129, ...

%e 0, 1, 1, 3, 3, 13, 13, 63, 63, ...

%e 0, 0, 1, 1, 5, 5, 25, 25, 129, ...

%e 0, 0, 0, 1, 1, 7, 7, 41, 41, ...

%e 0, 0, 0, 0, 1, 1, 9, 9, 61, ...

%e 0, 0, 0, 0, 0, 1, 1, 11, 11, ...

%e Triangle T(n,k) recurrence: 63 = T(6,3)= 25 + 13 + 25.

%e Triangle T(n,k) recurrence with A-sequence A112478: 63 = T(6,3) = 1*25 + 2*25 - 2*9 + 6*1 (T entries from row n=5 only).

%e From _Roger L. Bagula_, Dec 09 2008: (Start)

%e As a triangle this begins:

%e {1},

%e {1, 1},

%e {1, 3, 1},

%e {1, 5, 5, 1},

%e {1, 7, 13, 7, 1},

%e {1, 9, 25, 25, 9, 1},

%e {1, 11, 41, 63, 41, 11, 1},

%e {1, 13, 61, 129, 129, 61, 13, 1},

%e {1, 15, 85, 231, 321, 231, 85, 15, 1},

%e {1, 17, 113, 377, 681, 681, 377, 113, 17, 1},

%e {1, 19, 145, 575, 1289, 1683, 1289, 575, 145, 19, 1} ...

%e (End)

%e From _Philippe Deléham_, Mar 29 2012: (Start)

%e Subtriangle of the triangle given by (1, 0, 1, -1, 0, 0, 0, ...) DELTA (0, 1, 0, 0, 0, ...) where DELTA is the operator defined in A084938:

%e 1

%e 1, 0

%e 1, 1, 0

%e 1, 3, 1, 0

%e 1, 5, 5, 1, 0

%e 1, 7, 13, 7, 1, 0

%e 1, 9, 25, 25, 9, 1, 0

%e 1, 11, 41, 63, 41, 11, 1, 0 ...

%e Subtriangle of the triangle given by (0, 1, 0, 0, 0, ...) DELTA (1, 0, 1, -1, 0, 0, 0, ...) where DELTA is the operator defined in A084938:

%e 1

%e 0, 1

%e 0, 1, 1

%e 0, 1, 3, 1

%e 0, 1, 5, 5, 1

%e 0, 1, 7, 13, 7, 1

%e 0, 1, 9, 25, 25, 9, 1

%e 0, 1, 11, 41, 63, 41, 11, 1 ... (End)

%p A008288 := proc(n, k) option remember; if k = 0 then 1 elif n=k then 1 else A008288(n-1, k-1) + A008288(n-2, k-1) + A008288(n-1, k) fi; end: seq(seq(A008288(n,k),k=0..n), n=0..10);

%p P[0]:=1; P[1]:=x+1; for n from 2 to 12 do P[n]:=expand((x+1)*P[n-1]+x*P[n-2]); lprint(P[n]); lprint(seriestolist(series(P[n],x,200))); od:

%t Clear[a]; a[0] = {1}; a[1] = {1, 1}; a[n_] := a[n] = Join[{0}, a[n - 2], {0}] + Join[{0}, a[n - 1]] + Join[a[n - 1], {0}]; Table[a[n], {n, 0, 10}]; Flatten[%] (* _Roger L. Bagula_, Dec 09 2008 *)

%t (* Next, A008388 jointly generated with A035607 *)

%t u[1, x_] := 1; v[1, x_] := 1; z = 16;

%t u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];

%t v[n_, x_] := 2 x*u[n - 1, x] + v[n - 1, x];

%t Table[Expand[u[n, x]], {n, 1, z/2}]

%t Table[Expand[v[n, x]], {n, 1, z/2}]

%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

%t TableForm[cu]

%t Flatten[%] (* A008288 *)

%t Table[Expand[v[n, x]], {n, 1, z}]

%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

%t TableForm[cv]

%t Flatten[%] (* A035607 *)

%t (* _Clark Kimberling_, Mar 09 2012 *)

%t d[n_, k_] := Binomial[n+k, k]*Hypergeometric2F1[-k, -n, -n-k, -1]; A008288 = Flatten[Table[d[n-k, k], {n, 0, 12}, {k, 0, n}]] (* _Jean-François Alcover_, Apr 05 2012, after 3rd formula *)

%o (PARI) /* computation as lattice paths: */

%o /* same as in A092566, but last line (output) replaced by either of the following */

%o /* show as square array: */

%o M

%o /* show as triangle T(n-k,k): */

%o { for(n=0,N-1, for(k=0,n, print1(T(n-k,k),", "); ); print(); ); }

%o /* _Joerg Arndt_, Jul 01 2011 */

%o a008288 n k = a008288_tabl !! n !! k

%o a008288_row n = a008288_tabl !! n

%o a008288_tabl = map fst \$ iterate

%o (\(us, vs) -> (vs, zipWith (+) ([0] ++ us ++ [0]) \$

%o zipWith (+) ([0] ++ vs) (vs ++ [0]))) ([1], [1, 1])

%o -- _Reinhard Zumkeller_, Jul 21 2013

%o (Sage)

%o for k in range(8):

%o a = lambda n: hypergeometric([-n, -k], [1], 2)

%o print [simplify(a(n)) for n in range(11)] # _Peter Luschny_, Nov 19 2014

%Y Sums of antidiagonals = A000129 (Pell numbers), D(n, n) = A001850 (Delannoy numbers), (T(n, 3)) = A001845, (T(n, 4)) = A001846, etc. See also A027618. Rows and diagonals give A001844-A001850. Cf. A059446.

%Y See central Delannoy numbers A001850 for further information and references.

%Y Has same main diagonal as A064861. Different from A100936.

%Y Cf. A101164, A101167, A128966.

%Y Cf. A131887, A131935.

%Y Cf. A035607, A108625, A142979, A142992, A143007.

%Y Read mod small primes: A211312-A211315.

%Y Triangle sums (see the comments): A000129 (Row1); A056594 (Row2); A000073 (Kn11 & Kn21); A089068 (Kn12 & Kn22); A180668 (Kn13 & Kn23); A180669 (Kn14 & Kn24); A180670 (Kn15 & Kn25); A099463 (Kn3 & Kn4); A116404 (Fi1 & Fi2); A006498 (Ca1 & Ca2); A006498(3*n) (Ca3 & Ca4); A079972 (Gi1 & Gi2); A079972(4*n) (Gi3 & Gi4); A079973(3*n) (Ze1 & Ze2); A079973(2*n) (Ze3 & Ze4).

%Y Cf. A102413, A128966.

%Y (D(n,1)) = A005843

%K nonn,tabl,nice,easy,changed

%O 0,5

%A _N. J. A. Sloane_

%E Expanded description from _Clark Kimberling_, Jun 15 1997

%E Additional references from Sylviane R. Schwer (schwer(AT)lipn.univ-paris13.fr), Nov 28 2001

%E Changed the notation to make the formulas more precise. - _N. J. A. Sloane_, Jul 01 2002

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Last modified January 21 02:59 EST 2019. Contains 319344 sequences. (Running on oeis4.)