A008288 Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.
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
Offset: 0
Examples
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 A-sequence 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..n-k} A112478(j) * T(n-1, k-1+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)
References
- Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 593.
- Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.
- L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
- L. Moser and W. Zayachkowski, Lattice paths with diagonal steps, Scripta Mathematica, 26 (1963), 223-229.
- 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), 42-45.
- Said Amrouche and Hacène Belbachir, Asymmetric extension of Pascal-Dellanoy 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 Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, 9 (2006), #06.2.4.
- Paul Barry, Riordan-Bernstein Polynomials, Hankel Transforms and Somos Sequences, Journal of Integer Sequences, 15 (2012), #12.8.2.
- Paul Barry, On the Inverses of a Family of Pascal-Like Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.6.
- Paul Barry, The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1804.05027 [math.CO], 2018.
- Paul Barry, The Central Coefficients of a Family of Pascal-like Triangles and Colored Lattice Paths, J. Int. Seq., 22 (2019), #19.1.3.
- Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., 22 (2019), #19.5.8.
- Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 3, 29.
- Frédéric Bihan, Francisco Santos, and Pierre-Jean Spaenlehauer, A Polyhedral Method for Sparse Systems with many Positive Solutions, arXiv:1804.05683 [math.CO], 2018.
- Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, 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), 1-19.
- C. Carré, N. Debroux, M. Deneufchatel, J.-P. Dubernard et al., Dirichlet convolution and enumeration of pyramid polycubes, hal-00905889, 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), 2623-2628.
- Swee Hong Chan, Igor Pak, and Greta Panova, Log-concavity 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. 70-90 (see the table given on p. 76).
- Jerry Ray Dias, Properties and relationships of conjugated polyenes having a reciprocal eigenvalue spectrum - dendralene and radialene hydrocarbons, Croatica Chem. Acta, 77 (2004), 325-330.
- 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, Generalizations of Delannoy and cross polytope numbers, Fib. Q., Vol. 55, No. 4 (2017), pp. 356-366.
- Steven Edwards and William Griffiths, On Generalized Delannoy Numbers, J. Int. Seq., 23 (2020), #20.3.6.
- R. Feria-Puron, H. Perez-Roses, and J. Ryan, Searching for Large Circulant Graphs, arXiv:1503.07357 [math.CO], 2015.
- R. Feria-Purón, J. Ryan, and H. Pérez-Rosés, Searching for Large Multi-Loop Networks, Electronic Notes in Discrete Mathematics, 46 (2014), 233-240.
- Nate Harman, Andrew Snowden, and Noah Snyder, The Delannoy Category, arxiv:2211.15392 [math.RT], 2023.
- Rebecca Hartman-Baker, The Diffusion Equation Method for Global Optimization and Its Application to Magnetotelluric Geoprospecting, University of Illinois, Urbana-Champaign, 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, 1974-1975.
- Kentaro Ihara, Yayoi Nakamura, and Shuji Yamamoto, Interpolant of truncated multiple zeta functions, arXiv:2407.20509 [math.NT], 2024. See p. 21.
- Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
- Milan Janjić and B. Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013. - _N. J. A. Sloane_, Feb 13 2013
- Milan Janjić 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, Tables of Sequences Related to Delannoy Numbers and Cubic Lattice Coordination Numbers
- 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. 4-10.
- 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), 1-8. [Annotated scanned copy]
- Eon Lee, Andrés R. Vindas-Meléndez, and Zhi Wang, Generalized snake posets, order polytopes, and lattice-point enumeration, arXiv:2411.18695 [math.CO], 2024. See p. 15.
- Yi-Lin Lee, Off-diagonally symmetric domino tilings of the Aztec diamond of odd order, arXiv:2404.09057 [math.CO], 2024. See p. 20.
- Huyile Liang, Yanni Pei, and Yi Wang, Analytic combinatorics of coordination numbers of cubic lattices, arXiv:2302.11856 [math.CO], 2023. See p. 1.
- M. LLadser, Uniform formulas for coefficients of meromorphic functions, arXiv:math/0604152 [math.CO], 2006.
- E. Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 174.
- Matt Majic, Relationships between spherical and bispherical harmonics, and an electrostatic T-matrix 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), 589-598.
- Mirka Miller, Hebert Perez-Roses, and Joe Ryan, The Maximum Degree-and-Diameter-Bounded 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 Sai-nan Zheng, On the Total Positivity of Delannoy-Like Triangles, Journal of Integer Sequences, 20 (2017), #17.1.6.
- M. Norfleet, Characterization of second-order strong divisibility sequences of polynomials, The Fibonacci Quarterly, 43(2) (2005), 166-169.
- Richard L. Ollerton and Anthony G. Shannon, Some properties of generalized Pascal squares and triangles, Fib. Q., 36 (1998), 98-109. See Table 9.
- L. Pachter and B. Sturmfels, The mathematics of phylogenomics, arXiv:math/0409132 [math.ST], 2004-2005.
- 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 self-similarity structure generated by king's walk, Algebraic and topological methods in graph theory (Lake Bled, 1999). Discrete Math. 244(1-3) (2002), 423--433. 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), 75-89.
- Seunghyun Seo, The Catalan Threshold Arrangement, Journal of Integer Sequences, 20 (2017), #17.1.1.
- Yuriy Shablya, Combinatorial Generation Algorithms for Some Lattice Paths Using the Method Based on AND/OR Trees, Algorithms (2023) Vol. 16, No. 6, 266.
- 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), 277-279.
- Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.
- Yi Wang, Zheng Sai-Nan, and Chen Xi, Analytic aspects of Delannoy numbers, Discrete Mathematics 342(8) (2019), 2270-2277.
- Eric Weisstein's World of Mathematics, Delannoy Number.
- Meng-Han Wu, Henryk A. Witek, Rafał Podeszwa, Clar Covers and Zhang-Zhang Polynomials of Zigzag and Armchair Carbon Nanotubes, MATCH Commun. Math. Comput. Chem. (2025) Vol. 93, 415-462. See p. 445.
- Dmitry Zaitsev, k-neighborhood 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.
Crossrefs
Sums of antidiagonals: A000129 (Pell numbers).
Main diagonal: A001850 (central Delannoy numbers), which has further information and references.
Rows 0..10: A000012, A005408, A001844, A001845, A001846, A001847, A001848, A001849, A008417, A008419, A008421.
See also A027618.
Cf. A059446.
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).
Programs
-
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 -
Maple
A008288 := proc(n, k) option remember; if k = 0 then 1 elif n=k then 1 else procname(n-1, k-1) + procname(n-2, k-1) + procname(n-1, k) end if; end proc: seq(seq(A008288(n,k),k=0..n), n=0..10); # triangular indices n and k 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))); end do:
-
Mathematica
(* Next, A008288 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, -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 *)
-
Python
from functools import cache @cache def delannoy_row(n: int) -> list[int]: if n == 0: return [1] if n == 1: return [1, 1] rov = delannoy_row(n - 2) row = delannoy_row(n - 1) + [1] for k in range(n - 1, 0, -1): row[k] += row[k - 1] + rov[k - 1] return row for n in range(10): print(delannoy_row(n)) # Peter Luschny, Jul 30 2023 -
Sage
for k in range(8): # seen as an array, read row by row a = lambda n: hypergeometric([-n, -k], [1], 2) print([simplify(a(n)) for n in range(11)]) # Peter Luschny, Nov 19 2014
Formula
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).
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+k-d, 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(n-1, k-1) + T(n-2, k-1) + T(n-1, k), 0 < k < n and n > 1. - Reinhard Zumkeller, Dec 03 2004
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
T(n,k) = Sum_{j=0..n-k} C(k,j)*C(n-j,k). - Paul Barry, May 21 2006
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
Recurrence for triangle from A-sequence (see the Wolfdieter Lang comment above): T(n,k) = Sum_{j=0..n-k} A112478(j) * T(n-1, k-1+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 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.
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. (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(x-1) = (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(x-1), 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 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} (-1)^(k+1)/(k*D(n,k-1)*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(n-1,n-1)*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)
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,n-k;3) defines the table of asymmetric Delannoy numbers, essentially A049600. (End)
Seen as a triangle read by rows: T(n, k) = Hyper2F1([k-n, -k], [1], 2). - Peter Luschny, Aug 02 2014, Oct 13 2024.
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 n-th 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 n-th subdiagonal of triangle T(n,k) is the n-th 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(n-i, i) * binomial(n-2*i, k-i) 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(n-1,k). [Majic (2019), Eq. 22] - Matt Majic, Nov 24 2019
For i, j >= 1, D(i,j) = D(i,j-1) + 2*Sum_{k=0..i-1} D(k,j-1), or, because D(i,j) = D(j,i), D(i,j) = D(i-1,j) + 2*Sum_{k=0..j-1} D(i-1,k). - Shel Kaphan, Jan 01 2023
Sum_{k=0..n} T(n,k)^2 = A026933(n). - R. J. Mathar, Nov 07 2023
Let S(x) = (1 - x - (1 - 6*x + x^2)^(1/2))/(2*x) denote the g.f. of the sequence of large Schröder numbers A006318. Read as a lower triangular array, the signed n-th row polynomial R(n, -x) = 1/sqrt(1 - 6*x + x^2) *( 1/S(x)^(n+1) + (x*S(x))^(n+1) ). For example, R(4, -x) = 1 - 7*x + 13*x^2 - 7*x^3 + x^4 = 1/sqrt(1 - 6*x + x^2) * ( 1/S(x)^5 + (x*S(x))^5 ). Cf. A102413. - Peter Bala, Aug 01 2024
Extensions
Expanded description from Clark Kimberling, Jun 15 1997
Additional references from Sylviane R. Schwer (schwer(AT)lipn.univ-paris13.fr), Nov 28 2001
Changed the notation to make the formulas more precise. - N. J. A. Sloane, Jul 01 2002
Comments