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)lhsystems.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.

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?

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

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)lhsystems.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.

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).

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:

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.

Sequence in context: A103450 A128254 A026714 this_sequence A106597 A108359 A100936

Adjacent sequences: A008285 A008286 A008287 this_sequence A008289 A008290 A008291

nonn,tabl,nice,easy

njas

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. - njas, Jul 01, 2002

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