

A008288


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


125



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, 61, 129, 129, 61, 13, 1, 1, 15, 85, 231, 321, 231, 85, 15, 1, 1, 17, 113, 377, 681, 681, 377, 113, 17, 1, 1, 19, 145, 575, 1289, 1683, 1289, 575, 145, 19, 1, 1, 21, 181, 833, 2241, 3653, 3653
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OFFSET

0,5


COMMENTS

In the Formula section, some contributors use T(n,k) = D(nk, k) (for 0 <= k <= n), which is the triangular version of the square array (D(n,k): n,k >= 0). Conversely, D(n,k) = T(n+k,k) for n,k >= 0.  Petros Hadjicostas, Aug 05 2020
Also called the tribonacci triangle [Alladi and Hoggatt (1977)].  N. J. A. Sloane, Mar 23 2014
D(n,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 [Corrected by N. J. A. Sloane, May 30 2020]
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[n1] + x*P[n2].
D(n, k) is the number of kmatchings of a comblike 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 3matchings: four triples of three teeth and the three triples {Aa, Bb, CD}, {Aa, Dd, BC}, {Cc, Dd, AB}. Also D(3, 1)=7, the 1matchings of the same graph being the seven edges: {AB}, {BC}, {CD}, {Aa}, {Bb}, {Cc}, {Dd}.  Emeric Deutsch, Jul 01 2002
Sum of nth antidiagonal of the array D is A000129(n+1).  Reinhard Zumkeller, Dec 03 2004 [Edited by Petros Hadjicostas, Aug 05 2020 so that the counting of antidiagonals of D starts at n = 0. That is, the sum of the terms in the nth row of the triangles T is A000129(n+1).]
The Asequence for this Riordan type triangle (see one of Paul Barry's comments under Formula) is A112478 and the Zsequence the trivial: {1, 0, 0, 0, ...}. See the W. Lang link under A006232 for Sheffer a and zsequences where also Riordan A and Zsequences are explained. This leads to the recurrence for the triangle given below.  Wolfdieter Lang, Jan 21 2008
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
This sequence, 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(n1,x) + v(n1) and v(n,x) = 2*x*u(n1,x) + v(n1,x). See the Mathematica section.  Clark Kimberling, Mar 09 2012
Row n, for n > 0, of 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 nth convergent of the continued fraction [k, k, k, ...], where k = sqrt(x) + 1/sqrt(x); see A230000.  Clark Kimberling, Nov 13 2013
In an ndimensional 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 are called the von Neumann neighborhood of radius k. For k=1, see A005843.  Dmitry Zaitsev, Dec 10 2015
These numbers appear as the coefficients of series relating spherical and bispherical harmonics, in the solutions of Laplace's equation in 3D. [Majic 2019, Eq. 22]  Matt Majic, Nov 24 2019
From Peter Bala, Feb 19 2020: (Start)
The following remarks assume an offset of 1 in the row and column indices of the triangle.
The sequence of row polynomials T(n,x), beginning with T(1,x) = x, T(2,x) = x + x^2, T(3,x) = x + 3*x^2 + x^3, ..., is a strong divisibility sequence of polynomials in the ring Z[x]; that is, for all positive integers n and m, poly_gcd(T(n,x), T(m,x)) = T(gcd(n, m), x)  apply Norfleet (2005), Theorem 3. Consequently, the sequence (T(n,x): n >= 1) is a divisibility sequence in the polynomial ring Z[x]; that is, if n divides m then T(n,x) divides T(m,x) in Z[x].
Let S(x) = 1 + 2*x + 6*x^2 + 22*x^3 + ... denote the o.g.f. for the large Schröder numbers A006318. The power series (x*S(x))^n, n = 2, 3, 4, ..., can be expressed as a linear combination with polynomial coefficients of S(x) and 1: (x*S(x))^n = T(n1,x)  T(n,x)*S(x). The result can be extended to negative integer n if we define T(0,x) = 0 and T(n,x) = (1)^(n+1) * T(n,x)/x^n. Cf. A115139.
[In the previous two paragraphs, D(n,x) was replaced with T(n,x) because the contributor is referring to the rows of the triangle T(n,k), not the rows of the array D(n,k).  Petros Hadjicostas, Aug 05 2020] (End)
Named after the French amateur mathematician HenriAuguste Delannoy (18331915).  Amiram Eldar, Apr 15 2021
D(i,j) = D(j,i). With this and Dmitry Zaitsev's Dec 10 2015 comment, D(i,j) can be considered the number of points at L1 distance <= i in Z^j or the number of points at L1 distance <= j in Z^i from any given point. The rows and columns of D(i,j) are the crystal ball sequences on cubic lattices. See the first example below. The nth term in the kth crystal ball sequence can be considered the number of points at distance <= n from any point in a kdimensional cubic lattice, or the number of points at distance <= k from any point in an ndimensional cubic lattice.  Shel Kaphan, Jan 01 2023 and Jan 07 2023


REFERENCES

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 593.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
Steven Edwards and W. Griffiths, Generalizations of Delannoy and cross polytope numbers, Fib. Q., 55 (2017), 356366.
L. Moser and W. Zayachkowski, Lattice paths with diagonal steps, Scripta Mathematica, 26 (1963), 223229.
G. Picou, Note #2235, L'Intermédiaire des Mathématiciens, 8 (1901), page 281.  N. J. A. Sloane, Mar 02 2022
D. B. West, Combinatorial Mathematics, Cambridge, 2021, p. 28.


LINKS

T. D. Noe, Table of n, a(n) for n = 0..5150
K. Alladi and V. E. Hoggatt Jr., On tribonacci numbers and related functions, Fibonacci Quart. 15 (1977), 4245.
Said Amrouche and Hacène Belbachir, Asymmetric extension of PascalDellanoy triangles, arXiv:2001.11665 [math.CO], 2020.
J.M. Autebert et al., H.A. Delannoy et les oeuvres posthumes d'Édouard Lucas, Gazette des Mathématiciens  no 95, Jan 2003 (in French).
Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], 2017. See Sect. 2.3.
C. Banderier and S. Schwer, Why Delannoy numbers?, arXiv:math/0411128 [math.CO], 2004.
Paul Barry, On IntegerSequenceBased Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, 9 (2006), #06.2.4.
Paul Barry, RiordanBernstein Polynomials, Hankel Transforms and Somos Sequences, Journal of Integer Sequences, 15 (2012), #12.8.2.
Paul Barry, On the Inverses of a Family of PascalLike Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.6.
Paul Barry, The GammaVectors of Pascallike Triangles Defined by Riordan Arrays, arXiv:1804.05027 [math.CO], 2018.
Paul Barry, The Central Coefficients of a Family of Pascallike Triangles and Colored Lattice Paths, J. Int. Seq., 22 (2019), #19.1.3.
Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascallike Triangles, J. Int. Seq., 22 (2019), #19.5.8.
Frédéric Bihan, Francisco Santos, and PierreJean Spaenlehauer, A Polyhedral Method for Sparse Systems with many Positive Solutions, arXiv:1804.05683 [math.CO], 2018.
B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5648007388. English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 37.
D. Bump, K. Choi, P. Kurlberg and J. Vaaler, A local Riemann hypothesis, I, Math. Zeit. 233 (2000), 119.
C. Carré, N. Debroux, M. Deneufchatel, J.P. Dubernard et al., Dirichlet convolution and enumeration of pyramid polycubes, hal00905889, 2013.
C. Carre, N. Debroux, M. Deneufchatel, J.Ph. Dubernard, C. Hillariet, J.G. Luque, and O. Mallet, Enumeration of Polycubes and Dirichlet Convolutions, J. Int. Seq. 18 (2015), #15.11.4.
J. S. Caughman et al., A note on lattice chains and Delannoy numbers, Discrete Math., 308 (2008), 26232628.
Swee Hong Chan, Igor Pak, and Greta Panova, Logconcavity in planar random walks, arXiv:2106.10640 [math.CO], 2021.
H. Delannoy, Emploi de l'échiquier pour la résolution de certains problèmes de probabilités, Association Française pour l'Avancement des Sciences, 24th session, 1895, pp. 7090 (see the table given on p. 76).
J. R. Dias, Properties and relationships of conjugated polyenes having a reciprocal eigenvalue spectrum  dendralene and radialene hydrocarbons, Croatica Chem. Acta, 77 (2004), 325330.
M. Dziemianczuk, Generalizing Delannoy numbers via counting weighted lattice paths, INTEGERS, 13 (2013), #A54.
James East and Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.
Steven Edwards and William Griffiths, On Generalized Delannoy Numbers, J. Int. Seq., 23 (2020), #20.3.6.
R. FeriaPuron, H. PerezRoses, and J. Ryan, Searching for Large Circulant Graphs, arXiv:1503.07357 [math.CO], 2015.
R. FeriaPurón, J. Ryan, and H. PérezRosés, Searching for Large MultiLoop Networks, Electronic Notes in Discrete Mathematics, 46 (2014), 233240.
Rebecca HartmanBaker, The Diffusion Equation Method for Global Optimization and Its Application to Magnetotelluric Geoprospecting, University of Illinois, UrbanaChampaign, 2005.
G. Hetyei, Shifted Jacobi polynomials and Delannoy numbers, arXiv:0909.5512 [math.CO], 2009.
G. Hetyei, Links we almost missed between Delannoy numbers and Legendre polynomials.
V. E. Hoggatt, Jr., Letters to N. J. A. Sloane, 19741975.
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
M. Janjic and B. Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.  N. J. A. Sloane, Feb 13 2013
M. Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014), #14.3.5.
Svante Janson, Patterns in random permutations avoiding some sets of multiple patterns, arXiv:1804.06071 [math.PR], 2018.
Shel Kaphan, Illustration of a recurrence relation on the Delannoy numbers and their connection with geometry.
G. Kreweras, Sur les hiérarchies de segments, Cahiers Bureau Universitaire Recherche Opérationnelle, # 20, Inst. Statistiques, Univ. Paris, 1973, pp. 410.
G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
G. Kreweras, Aires des chemins surdiagonaux et application à un problème économique, Cahiers du Bureau universitaire de recherche opérationnelle Série Recherche 24 (1976), 18. [Annotated scanned copy]
M. LLadser, Uniform formulas for coefficients of meromorphic functions, arXiv:math/0604152 [math.CO], 2006.
E. Lucas, Théorie des Nombres. GauthierVillars, Paris, 1891, Vol. 1, p. 174.
Matt Majic, Relationships between spherical and bispherical harmonics, and an electrostatic Tmatrix for dimers, preprint, 2019, DOI:10.13140/RG.2.2.21203.12320.
J. W. Meijer, Famous numbers on a chessboard, Acta Nova, 4(4) (2010), 589598.
Mirka Miller, Hebert PerezRoses, and Joe Ryan, The Maximum DegreeandDiameterBounded Subgraph in the Mesh, arXiv:1203.4069 [math.CO], 2012.
Alejandro H. Morales, Igor Pak, and Greta Panova, Hook formulas for skew shapes IV. Increasing tableaux and factorial Grothendieck polynomials, arXiv:2108.10140 [math.CO], 2021.
Lili Mu and Sainan Zheng, On the Total Positivity of DelannoyLike Triangles, Journal of Integer Sequences, 20 (2017), #17.1.6.
M. Norfleet, Characterization of secondorder strong divisibility sequences of polynomials, The Fibonacci Quarterly, 43(2) (2005), 166169.
Richard L. Ollerton and Anthony G. Shannon, Some properties of generalized Pascal squares and triangles, Fib. Q., 36 (1998), 98109. See Table 9.
L. Pachter and B. Sturmfels, The mathematics of phylogenomics, arXiv:math/0409132 [math.ST], 20042005.
R. Pemantle and M. C. Wilson, Asymptotics of multivariate sequences, I: smooth points of the singular variety, arXiv:math/0003192 [math.CO], 2000.
J. L. Ramirez and V. F. Sirvent, Incomplete Tribonacci Numbers and Polynomials, Journal of Integer Sequences, 17 (2014), #14.4.2. See Table 1.  N. J. A. Sloane, Mar 23 2014
Marko Razpet, A selfsimilarity structure generated by king's walk, Algebraic and topological methods in graph theory (Lake Bled, 1999). Discrete Math. 244(13) (2002), 423433. MR1844050 (2002k:05022).
Shiva Samieinia, Digital straight line segments and curves, Licentiate Thesis. Stockholm University, Department of Mathematics, Report 2007:6.
Shiva Samieinia, The number of continuous curves in digital geometry, Port. Math. 67(1) (2010), 7589.
Seunghyun Seo, The Catalan Threshold Arrangement, Journal of Integer Sequences, 20 (2017), #17.1.1.
M. Shattuck, Combinatorial identities for incomplete tribonacci polynomials, arXiv:1406.2755 [math.CO], 2014.
R. G. Stanton and D. D. Cowan, Note on a "square" functional equation, SIAM Rev., 12 (1970), 277279.
Luis VerdeStar A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.
Yi Wang, Zheng SaiNan, and Chen Xi, Analytic aspects of Delannoy numbers, Discrete Mathematics 342(8) (2019), 22702277.
Eric Weisstein's World of Mathematics, Delannoy Number.
Dmitry Zaitsev, kneighborhood for Cellular Automata, arXiv:1605.08870 [cs.DM], 2016.
Liang Zhao and Fengyao Yan, Note on Total Positivity for a Class of Recursive Matrices, Journal of Integer Sequences, 19 (2016), #16.6.5.


FORMULA

D(n, 0) = 1 = D(0, n) for n >= 0; D(n, k) = D(n, k1) + D(n1, k1) + D(n1, k).
Bivariate o.g.f.: Sum_{n >= 0, k >= 0} D(n, k)*x^n*y^k = 1/(1  x  y  x*y).
D(n, k) = Sum_{d = 0..min(n,k)} binomial(k, d)*binomial(n+kd, k) = Sum_{d=0..min(n,k)} 2^d*binomial(n, d)*binomial(k, d). [Edited by Petros Hadjicostas, Aug 05 2020]
Seen as a triangle read by rows: T(n, 0) = T(n, n) = 1 for n >= 0 and T(n, k) = T(n1, k1) + T(n2, k1) + T(n1, k), 0 < k < n and n > 1.  Reinhard Zumkeller, Dec 03 2004
Read as a number triangle, this is the Riordan array (1/(1x), x(1+x)/(1x)) with T(n, k) = Sum_{j=0..nk} C(nk, j) * C(k, j) * 2^j.  Paul Barry, Jul 18 2005
T(n,k) = Sum_{j=0..nk} C(k,j)*C(nj,k).  Paul Barry, May 21 2006
Let y^k(n) be the number of Khalimskycontinuous functions f from [0,n1] to Z such that f(0) = 0 and f(n1) = k. Then y^k(n) = D(i,j) for i = (1/2)*(n1k) and j = (1/2)*(n1+k) where n1+k belongs to 2Z.  Shiva Samieinia (shiva(AT)math.su.se), Oct 08 2007
Recurrence for triangle from Asequence (see the Wolfdieter Lang comment above): T(n,k) = Sum_{j=0..nk} A112478(j) * T(n1, k1+j), n >= 1, k >= 1. [For k > n, the sum is empty, in which case T(n,k) = 0.]
From Peter Bala, Jul 17 2008: (Start)
The nth 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.
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 nth row of the square array, is the Ehrhart polynomial of the ndimensional cross polytope (the hyperoctahedron) [Bump et al. (2000), Theorem 6].
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.
The reciprocity law p_n(m) = p_m(n) reflects the symmetry of the table.
The polynomial p_n(x) is the unique polynomial solution of the difference equation (x+1)*f(x+1)  x*f(x1) = (2*n+1)*f(x), normalized so that f(0) = 1.
These polynomials have their zeros on the vertical line Re x = 1/2 in the complex plane; that is, the polynomials p_n(x1), n = 1,2,3,..., satisfy a Riemann hypothesis [Bump et al. (2000), 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 + ... .
The square array of Delannoy numbers has a close connection with the constant log(2). The entries in the nth 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} (1)^(k+1)/(k*D(n,k1)*D(n,k)). [T(n,k) was replaced with D(n,k) in the formula to agree with the beginning of the paragraph.  Petros Hadjicostas, Aug 05 2020]
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.
Also the main diagonal entries (the central Delannoy numbers) give the series acceleration formula Sum_{n>=1} 1/(n*D(n1,n1)*D(n,n)) = (1/2)*log(2), a result due to Burnside. [T(n,n) was replaced here with D(n,n) to agree with the previous paragraphs.  Petros Hadjicostas, Aug 05 2020]
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)
From Peter Bala, Oct 28 2008: (Start)
Hilbert transform of Pascal's triangle A007318 (see A145905 for the definition of this term).
D(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,nk;3) defines the table of asymmetric Delannoy numbers, essentially A049600. (End)
Seen as a triangle read by rows: T(n,k) = (1)^(nk) * Hyper2F1([n+k, k+1], [1], 2) for 0 <= k <= n.  Peter Luschny, Aug 02 2014
From Peter Bala, Jun 25 2015: (Start)
O.g.f. for triangle T(n,k): 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 + ....
1 + z*d/dz(A(z,t))/A(z,t) is the o.g.f. for A102413. (End)
E.g.f. for the nth subdiagonal of T(n,k), n >= 0, 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 [The nth subdiagonal of triangle T(n,k) is the nth row of array D(n,k).]
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
Seen as a triangle read by rows: T(n,k) = Sum_{i=0..k} binomial(ni, i) * binomial(n2*i, ki) for 0 <= k <= n.  Werner Schulte, Jan 09 2019
Univariate generating function: Sum_{k >= 0} D(n,k)*z^k = (1 + z)^n/(1  z)^(n+1). [Dziemianczuk (2013), Eq. 5.3]  Matt Majic, Nov 24 2019
(n+1)*D(n+1,k) = (2*k+1)*D(n,k) + n*D(n1,k). [Majic (2019), Eq. 22]  Matt Majic, Nov 24 2019
For i, j >= 1, D(i,j) = D(i,j1) + 2*Sum_{k=0..i1} D(k,j1), or, because D(i,j) = D(j,i), D(i,j) = D(i1,j) + 2*Sum_{k=0..j1} D(i1,k).  Shel Kaphan, Jan 01 2023


EXAMPLE

The square array D(i,j) (i >= 0, j >= 0) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... = A000012
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, ... = A005408
1, 5, 13, 25, 41, 61, 85, 113, 145, 181, ... = A001844
1, 7, 25, 63, 129, 231, 377, 575, 833, 1159, ... = A001845
1, 9, 41, 129, 321, 681, 1289, 2241, 3649, 5641, ... = A001846
...
For D(2,5) = 61, which is seen above in the row labeled A001844, we calculate the sum (9 + 11 + 41) of the 3 nearest terms above and/or to the left.  Peter Munn, Jan 01 2023
D(2,5) = 61 can also be obtained from the row labeled A005408 using a recurrence mentioned in the formula section: D(2,5) = D(1,5) + 2*Sum_{k=0..4} D(1,k), so D(2,5) = 11 + 2*(1+3+5+7+9) = 11 + 2*25.  Shel Kaphan, Jan 01 2023
As a triangular array (on its side) this begins:
0, 0, 0, 0, 1, 0, 11, 0, ...
0, 0, 0, 1, 0, 9, 0, 61, ...
0, 0, 1, 0, 7, 0, 41, 0, ...
0, 1, 0, 5, 0, 25, 0, 129, ...
1, 0, 3, 0, 13, 0, 63, 0, ...
0, 1, 0, 5, 0, 25, 0, 129, ...
0, 0, 1, 0, 7, 0, 41, 0, ...
0, 0, 0, 1, 0, 9, 0, 61, ...
0, 0, 0, 0, 1, 0, 11, 0, ...
[Edited by Shel Kaphan, Jan 01 2023]
From Roger L. Bagula, Dec 09 2008: (Start)
As a triangle T(n,k) (with rows n >= 0 and columns k = 0..n), this begins:
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, 61, 129, 129, 61, 13, 1;
1, 15, 85, 231, 321, 231, 85, 15, 1;
1, 17, 113, 377, 681, 681, 377, 113, 17, 1;
1, 19, 145, 575, 1289, 1683, 1289, 575, 145, 19, 1;
... (End)
Triangle T(n,k) recurrence: 63 = T(6,3) = 25 + 13 + 25 = T(5,2) + T(4,2) + T(5,3).
Triangle T(n,k) recurrence with Asequence A112478: 63 = T(6,3) = 1*25 + 2*25  2*9 + 6*1 (T entries from row n = 5 only). [Here the formula T(n,k) = Sum_{j=0..nk} A112478(j) * T(n1, k1+j) is used with n = 6 and k = 3; i.e., T(6,3) = Sum_{j=0..3} A111478(j) * T(5, 2+j).  Petros Hadjicostas, Aug 05 2020]
From Philippe Deléham, Mar 29 2012: (Start)
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:
1;
1, 0;
1, 1, 0;
1, 3, 1, 0;
1, 5, 5, 1, 0;
1, 7, 13, 7, 1, 0;
1, 9, 25, 25, 9, 1, 0;
1, 11, 41, 63, 41, 11, 1, 0;
...
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:
1;
0, 1;
0, 1, 1;
0, 1, 3, 1;
0, 1, 5, 5, 1;
0, 1, 7, 13, 7, 1;
0, 1, 9, 25, 25, 9, 1;
0, 1, 11, 41, 63, 41, 11, 1;
... (End)


MAPLE

A008288 := proc(n, k) option remember; if k = 0 then 1 elif n=k then 1 else A008288(n1, k1) + A008288(n2, k1) + A008288(n1, k) fi; end: seq(seq(A008288(n, k), k=0..n), n=0..10);
P[0]:=1; P[1]:=x+1; for n from 2 to 12 do P[n]:=expand((x+1)*P[n1]+x*P[n2]); lprint(P[n]); lprint(seriestolist(series(P[n], x, 200))); od:


MATHEMATICA

(* Next, A008388 jointly generated with A035607 *)
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n  1, x] + v[n  1, x];
v[n_, x_] := 2 x*u[n  1, x] + v[n  1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%] (* A008288 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A035607 *)
(* Clark Kimberling, Mar 09 2012 *)
d[n_, k_] := Binomial[n+k, k]*Hypergeometric2F1[k, n, nk, 1]; A008288 = Flatten[Table[d[nk, k], {n, 0, 12}, {k, 0, n}]] (* JeanFrançois Alcover, Apr 05 2012, after 3rd formula *)


PROG

(Haskell)
a008288 n k = a008288_tabl !! n !! k
a008288_row n = a008288_tabl !! n
a008288_tabl = map fst $ iterate
(\(us, vs) > (vs, zipWith (+) ([0] ++ us ++ [0]) $
zipWith (+) ([0] ++ vs) (vs ++ [0]))) ([1], [1, 1])
 Reinhard Zumkeller, Jul 21 2013
(Sage)
for k in range(8):
a = lambda n: hypergeometric([n, k], [1], 2)
print([simplify(a(n)) for n in range(11)]) # Peter Luschny, Nov 19 2014


CROSSREFS

Sums of antidiagonals: A000129 (Pell numbers).
Main diagonal: A001850 (central Delannoy numbers), which has further information and references.
A002002, A026002, and A190666 are +kdiagonals for k=1, 2, 3 resp.  Shel Kaphan, Jan 01 2023
Rows 0..10: A000012, A005408, A001844, A001845, A001846, A001847, A001848, A001849, A008417, A008419, A008421.
See also A027618.
Cf. A059446.
Has same main diagonal as A064861. Different from A100936.
Cf. A101164, A101167, A128966, A131887, A131935.
Cf. A035607, A108625, A142979, A142992, A143007.
Read mod small primes: A211312, A211313, A211314, A211315.
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).
Cf. A102413, A128966. (D(n,1)) = A005843. Cf. A115139.
Sequence in context: A128254 A277930 A026714 * A238339 A302997 A326792
Adjacent sequences: A008285 A008286 A008287 * A008289 A008290 A008291


KEYWORD

nonn,tabl,nice,easy


AUTHOR

N. J. A. Sloane


EXTENSIONS

Expanded description from Clark Kimberling, Jun 15 1997
Additional references from Sylviane R. Schwer (schwer(AT)lipn.univparis13.fr), Nov 28 2001
Changed the notation to make the formulas more precise.  N. J. A. Sloane, Jul 01 2002


STATUS

approved



