0,5

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].

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 (deutsch(AT)duke.poly.edu), Jul 01 2002

Sum of n-th row = A000129(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 03 2004

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. W. Lang, Jan 21 2008.

Row sums are A000129. - L. Bagula (rlbagulatftn(AT)yahoo.com), Dec 09 2008)

Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (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.

J. S. Caughman et al., A note on lattice chains and Delannoy numbers, Discrete Math., 308 (2008), 2623-2628.

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

H. Delannoy. Emploi de l'echiquier pour la resolution de certains problemes de probabilites, Association Francaise pour l'Avancement des Sciences, 18-th session, 1895. pp. 70-90 (table given on pp. 76)

J. R. Dias, Properties and relationships of conjugated polyenes having a reciprocal eigenvalue spectrum - dendralene and radialene hydrocarbons, Croatica Chem. Acta, 77 (2004), 325-330.

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

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

R. G. Stanton and D. D. Cowan, Note on a "square" functional equation, SIAM Rev., 12 (1970), 277-279.

T. D. Noe, Table of n, a(n) for n=0..5150

C. Banderier and S. Schwer, Why Delannoy numbers?

D. Bump, K. Choi, P. Kurlberg and J. Vaaler, A local Riemann hypothesis, I, Math. Zeit. 233, (2000), 1-19.

Rebecca Hartman-Baker, The Diffusion Equation Method for Global Optimization and Its Application to Magnetotelluric Geoprospecting

G. Hetyei, Shifted Jacobi polynomials and Delannoy numbers [From Peter Bala (pbala(AT)toucansurf.com), Oct 28 2008]

G. Hetyei, Links we almost missed between Delannoy numbers and Legendre polynomials [From Peter Bala (pbala(AT)toucansurf.com), Nov 10 2008]

M. LLadser, Uniform formulae for coefficients of meromorphic functions

E. Lucas, Th\'{e}orie des Nombres. Gauthier-Villars, Paris, 1891, Vol. 1, p. 174.

L. Pachter and B. Sturmfels, The mathematics of phylogenomics

R. Pemantle and M. C. Wilson, Asymptotics of multivariate sequences, I: smooth points of the singular variety

Shiva Samieinia, Home Page.

Eric Weisstein's World of Mathematics, Delannoy Number

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).

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

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

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 (reinhard.zumkeller(AT)gmail.com), 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 (pbarry(AT)wit.ie), Jul 18 2005

T(n,k)=sum{j=0..n-k, C(k,j)C(n-j,k)}; - Paul Barry (pbarry(AT)wit.ie), 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 W. Lang comment above): T(n,k) = sum(A112478(j)*T(n-1,k-1+j),j=0..n-k), n>=1, k>=1.

Comments from Peter Bala (pbala(AT)toucansurf.com), 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., 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), normalised 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., 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..inf} (-1)^(k+1)/(k*T(n,k-1)*T(n,k)).

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..inf} 1/(n*T(n-1,n-1)*T(n,n)) = 1/2*log(2), a result due to Burnside.

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)

Contribution from Peter Bala (pbala(AT)toucansurf.com), Oct 28 2008: (Start)

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

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)

a(n) = Join[{0}, a(n - 2), {0}] + Join[{0}, a(n - 1)] + Join[a(n - 1), {0}]. [From Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Dec 09 2008]

Square array D(i,j) begins:

1 1 1 1 1 1 1 1 ...

1 3 5 7 9 11 13 ...

1 5 13 25 41 61 ...

1 7 25 63 129 231 ...

1 9 41 129 321 681 ...

...

As a triangular array (on its side) this begins

0,0,0,0,0,1,1,11,11 ...

0,0,0,0,1,1,9,9,61 ...

0,0,0,1,1,7,7,41,41 ...

0,0,1,1,5,5,25,25,129 ...

0,1,1,3,3,13,13,63,63 ...

0,0,1,1,5,5,25,25,129 ...

0,0,0,1,1,7,7,41,41 ...

0,0,0,0,1,1,9,9,61 ...

0,0,0,0,0,1,1,11,11 ...

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

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).

Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Dec 09 2008: (Start) As a triangle 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)

...

A008288 := proc(n, k) option remember; if n = 1 then 1; elif k = 1 then 1; else A008288(n-1, k-1)+A008288(n, k-1)+A008288(n-1, k); fi; end;

read transforms; SERIES2(1/(1-x-y-x*y), x, y, 12): SERIES2TOLIST(%, x, y, 12);

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:

Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Dec 09 2008: (Start)

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[%] (End)

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.

See central Delannoy numbers A001850 for further information and references.

Has same main diagonal as A064861. Different from A100936.

Cf. A101164, A101167, A128966.

Cf. A131887, A131935.

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

Sequence in context: A103450 A128254 A026714 this_sequence A144461 A106597 A108359

Adjacent sequences: A008285 A008286 A008287 this_sequence A008289 A008290 A008291

nonn,tabl,nice,easy

N. J. A. Sloane (njas(AT)research.att.com).

Expanded description from Clark Kimberling 6/97. Additional references from Sylviane R. Schwer (schwer(AT)lipn.univ-paris13.fr), Nov 28 2001.

I have changed the notation to make the formulae more precise. - N. J. A. Sloane (njas(AT)research.att.com), Jul 01, 2002

More terms from Shiva Samieinia (shiva(AT)math.su.se), Oct 08 2007