%I A008288
%S A008288 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,
%T A008288 129,129,61,13,1,1,15,85,231,321,231,85,15,1,1,17,113,377,681,681,377,
113,
%U A008288 17,1,1,19,145,575,1289,1683,1289,575,145,19,1,1,21,181,833,2241,3653,
3653
%N A008288 Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.
%C A008288 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 A008288 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
%C A008288 Sum of n-th row = A000129(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Dec 03 2004
%C A008288 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.
%C A008288 Row sums are A000129. - L. Bagula (rlbagulatftn(AT)yahoo.com), Dec 09
2008)
%D A008288 Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal
Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
%D A008288 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.
%D A008288 J. S. Caughman et al., A note on lattice chains and Delannoy numbers,
Discrete Math., 308 (2008), 2623-2628.
%D A008288 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
%D A008288 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)
%D A008288 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.
%D A008288 L. Moser and W. Zayachkowski, Lattice paths with diagonal steps, Scripta
Mathematica, 26 (1963) 223-229.
%D A008288 Shiva Samieinia, Digital straight line segments and curves. Licentiate
Thesis. Stockholm University, Department of Mathematics, Report 2007:6.
%D A008288 R. G. Stanton and D. D. Cowan, Note on a "square" functional equation,
SIAM Rev., 12 (1970), 277-279.
%H A008288 T. D. Noe, <a href="b008288.txt">Table of n, a(n) for n=0..5150</a>
%H A008288 C. Banderier and S. Schwer, <a href="http://arXiv.org/abs/math.CO/0411128">
Why Delannoy numbers?</a>
%H A008288 D. Bump, K. Choi, P. Kurlberg and J. Vaaler, <a href="http://citeseer.ist.psu.edu/
175986.html">A local Riemann hypothesis, I</a>, Math. Zeit. 233,
(2000), 1-19.
%H A008288 Rebecca Hartman-Baker, <a href="http://www.cs.uiuc.edu/research/techreports.php?report=UIUCDCS-R-2005-2578">
The Diffusion Equation Method for Global Optimization and Its Application
to Magnetotelluric Geoprospecting</a>
%H A008288 G. Hetyei, <a href="http://www.math.uncc.edu/preprint/2008/2008_08.pdf">
Shifted Jacobi polynomials and Delannoy numbers</a> [From Peter Bala
(pbala(AT)toucansurf.com), Oct 28 2008]
%H A008288 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> [From Peter Bala (pbala(AT)toucansurf.com),
Nov 10 2008]
%H A008288 M. LLadser, <a href="http://arXiv.org/abs/math.CO/0604152">Uniform formulae
for coefficients of meromorphic functions</a>
%H A008288 E. Lucas, <a href="http://gallica.bnf.fr/scripts/ConsultationTout.exe?E=0&O=N029021">
Th\'{e}orie des Nombres</a>. Gauthier-Villars, Paris, 1891, Vol.
1, p. 174.
%H A008288 L. Pachter and B. Sturmfels, <a href="http://arXiv.org/abs/math.ST/0409132">
The mathematics of phylogenomics</a>
%H A008288 R. Pemantle and M. C. Wilson, <a href="http://arXiv.org/abs/math.CO/0003192">
Asymptotics of multivariate sequences, I: smooth points of the singular
variety</a>
%H A008288 Shiva Samieinia, <a href="http://www.math.su.se/reports/2007/6/">Home
Page</a>.
%H A008288 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
DelannoyNumber.html">Delannoy Number</a>
%F A008288 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 A008288 Sum_{n >= 0, k >= 0} D(n, k)*x^n*y^k = 1/(1-x-y-x*y).
%F A008288 D(n, k) = Sum_{d} binomial(k, d)*binomial(n+k-d, k) = Sum_{d} 2^d*binomial(n,
d)*binomial(k, d).
%F A008288 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
%F A008288 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
%F A008288 T(n,k)=sum{j=0..n-k, C(k,j)C(n-j,k)}; - Paul Barry (pbarry(AT)wit.ie),
May 21 2006
%F A008288 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 A008288 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 A008288 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.
%F A008288 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 A008288 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 A008288 The reciprocity law p_n(m) = p_m(n) reflects the symmetry of the table.
%F A008288 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.
%F A008288 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 A008288 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 A008288 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 A008288 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 A008288 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 A008288 Contribution from Peter Bala (pbala(AT)toucansurf.com), Oct 28 2008:
(Start)
%F A008288 Hilbert transform of Pascal's triangle A007318 (see A145905 for the definition
of this term).
%F A008288 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.
%F A008288 (End)
%F A008288 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]
%e A008288 Square array D(i,j) begins:
%e A008288 1 1 1 1 1 1 1 1 ...
%e A008288 1 3 5 7 9 11 13 ...
%e A008288 1 5 13 25 41 61 ...
%e A008288 1 7 25 63 129 231 ...
%e A008288 1 9 41 129 321 681 ...
%e A008288 ...
%e A008288 As a triangular array (on its side) this begins
%e A008288 0,0,0,0,0,1,1,11,11 ...
%e A008288 0,0,0,0,1,1,9,9,61 ...
%e A008288 0,0,0,1,1,7,7,41,41 ...
%e A008288 0,0,1,1,5,5,25,25,129 ...
%e A008288 0,1,1,3,3,13,13,63,63 ...
%e A008288 0,0,1,1,5,5,25,25,129 ...
%e A008288 0,0,0,1,1,7,7,41,41 ...
%e A008288 0,0,0,0,1,1,9,9,61 ...
%e A008288 0,0,0,0,0,1,1,11,11 ...
%e A008288 Triangle T(n,k) recurrence: 63 = T(6,3)= 25 +13 +25.
%e A008288 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 A008288 Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Dec 09
2008: (Start) As a triangle this begins:
%e A008288 {1},
%e A008288 {1, 1},
%e A008288 {1, 3, 1},
%e A008288 {1, 5, 5, 1},
%e A008288 {1, 7, 13, 7, 1},
%e A008288 {1, 9, 25, 25, 9, 1},
%e A008288 {1, 11, 41, 63, 41, 11, 1},
%e A008288 {1, 13, 61, 129, 129, 61, 13, 1},
%e A008288 {1, 15, 85, 231, 321, 231, 85, 15, 1},
%e A008288 {1, 17, 113, 377, 681, 681, 377, 113, 17, 1},
%e A008288 {1, 19, 145, 575, 1289, 1683, 1289, 575, 145, 19, 1} (End)
%e A008288 ...
%p A008288 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;
%p A008288 read transforms; SERIES2(1/(1-x-y-x*y),x,y,12): SERIES2TOLIST(%,x,y,12);
%p A008288 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 A008288 Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Dec 09
2008: (Start)
%t A008288 Clear[a]; a[0] = {1}; a[1] = {1, 1};
%t A008288 a[n_] := a[n] = Join[{0}, a[n - 2], {0}] + Join[{0}, a[n - 1]] + Join[a[n
- 1], {0}];
%t A008288 Table[a[n], {n, 0, 10}]; Flatten[%] (End)
%Y A008288 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 A008288 See central Delannoy numbers A001850 for further information and references.
%Y A008288 Has same main diagonal as A064861. Different from A100936.
%Y A008288 Cf. A101164, A101167, A128966.
%Y A008288 Cf. A131887, A131935.
%Y A008288 Cf. A035607, A108625, A142979, A142992, A143007.
%Y A008288 Sequence in context: A103450 A128254 A026714 this_sequence A144461 A106597
A108359
%Y A008288 Adjacent sequences: A008285 A008286 A008287 this_sequence A008289 A008290
A008291
%K A008288 nonn,tabl,nice,easy
%O A008288 0,5
%A A008288 N. J. A. Sloane (njas(AT)research.att.com).
%E A008288 Expanded description from Clark Kimberling 6/97. Additional references
from Sylviane R. Schwer (schwer(AT)lipn.univ-paris13.fr), Nov 28
2001.
%E A008288 I have changed the notation to make the formulae more precise. - N. J.
A. Sloane (njas(AT)research.att.com), Jul 01, 2002
%E A008288 More terms from Shiva Samieinia (shiva(AT)math.su.se), Oct 08 2007
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