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A035607 Table a(d,m) of number of points of L1 norm m in cubic lattice Z^d, read by antidiagonals (d>=1, m >= 0). 18
1, 1, 2, 1, 4, 2, 1, 6, 8, 2, 1, 8, 18, 12, 2, 1, 10, 32, 38, 16, 2, 1, 12, 50, 88, 66, 20, 2, 1, 14, 72, 170, 192, 102, 24, 2, 1, 16, 98, 292, 450, 360, 146, 28, 2, 1, 18, 128, 462, 912, 1002, 608, 198, 32, 2, 1, 20, 162, 688, 1666, 2364, 1970, 952, 258, 36, 2, 1, 22, 200, 978, 2816 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Table also gives coordination sequences of same lattices.

Rows sums are given by A001333. Rising and falling diagonals are the tribonacci numbers A000213, A001590. - Paul Barry, Feb 13 2003

a(d,m) also gives the number of ways to choose m squares from a 2 X (d-1) grid so that no two squares in the selection are (horizontally or vertically) adjacent. - Jacob A. Siehler, May 13 2006

Mirror image of triangle A113413. - Philippe Deléham, Oct 15 2006

The Ca1 sums lead to A126116 and the Ca2 sums lead to A070550, see A180662 for the definitions of these triangle sums. - Johannes W. Meijer, Aug 05 2011

A035607 is jointly generated with the Delannoy triangle A008288 as an array of coefficients of polynomials v(n,x):  initially, u(1,x) = v(1,x) = 1; for n > 1, u(n,x) = x*u(n-1,x) + (x+1)*v(n-1) and v(n,x) = 2x*u(n-1,x) + v(n-1,x).  See the Mathematica section. - Clark Kimberling, Mar 05 2012

T(2*n,n) = A050146(n+1). - Reinhard Zumkeller, Jul 20 2013

Also, the polynomial v(n,x) above is x + (x + 1)f(n-1,x), where f(0,x) = 1. - Clark Kimberling, Oct 24 2014

LINKS

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (Abstract, pdf, ps).

M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550, 2013. - From N. J. A. Sloane, Feb 13 2013

Emanuele Munarini, Combinatorial properties of the antichains of a garland, Integers, 9 (2009), 353-374.

Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.

J. Siehler, Adjacency-free selections from a 2xN grid (Mathematica notebook)

FORMULA

From Johannes W. Meijer, Aug 05 2011: (Start)

f(d,m) = sum(binomial(floor((d-1+j)/2), d-m-1)*binomial(d-m-1, floor((d-1-j)/2)), j=0..d-1), d>=1 and 0<=m<=d-1.

f(d,m) = f(d-1,m-1) + f(d-1,m) + f(d-2,m-1) (d>=3 and 1 <= m <= d-1) with f(d,0) = 1 (d>=1) and f(d,d-1) = 2 (d>=2). (End)

From Roger Cuculière, Apr 10 2006: (Start)

The generating function G(x,y) of this double sequence is the sum of a(n,p)*x^n*y^p, n=1..infty, p=0..infty, which is G(x,y)=x*(1+y)/(1-x-y-(x*y)).

The horizontal generating function H_n(y), which generates the rows of the table: (1, 2, 2, 2, 2,...), (1, 4, 8, 12, 16, ...), (1, 6, 18, 38, 66, ...), is the sum of a(n,p)*y^p, p=0..infty, for each fixed n. This is H_n(y)=((1+y)^n)/((1-y)^n)).

The vertical generating function V_p(x), which generates the columns of the table: (1, 1, 1, 1, 1, ...}, (2, 4, 6, 8, 10, ...), (2, 8, 18, 32, 50, ...), is the sum of a(n,p)*x^n, n=1..infty, for each fixed p. This is V_p(x)=2*((1+x)^(p-1))/((1-x)^(p+1)) for p>=1 and V_0(x)=x/(1-x). (End)

G.f.: (1+x)/(1-x-x*y-x^2*y). - Vladeta Jovovic, Apr 02 2002 (But see previous lines!)

Seen as a triangle read by rows: T(n,0) = 1, for n > 1: T(n,n) = 2, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-1), 0 < k < n. - Reinhard Zumkeller, Jul 20 2013

EXAMPLE

As a triangle of coefficients in polynomials v(n,x) in Comments, the first 6 rows are

1

1      2

1      4      2

1      6      8       2

1      8      18      12     2

1      10     32      38     16    2

- Clark Kimberling, Oct 24 2014

MAPLE

A035607 := proc(d, m) local j: add(binomial(floor((d-1+j)/2), d-m-1)*binomial(d-m-1, floor((d-1-j)/2)), j=0..d-1) end: seq(seq(A035607(d, m), m=0..d-1), d=1..11); # d=dimension, m=norm [Johannes W. Meijer, Aug 05 2011]

MATHEMATICA

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 *)

PROG

(Haskell)

a035607 n k = a035607_tabl !! n !! k

a035607_row n = a035607_tabl !! n

a035607_tabl = map fst $ iterate

   (\(us, vs) -> (vs, zipWith (+) ([0] ++ us ++ [0]) $

                      zipWith (+) ([0] ++ vs) (vs ++ [0]))) ([1], [1, 2])

-- Reinhard Zumkeller, Jul 20 2013

CROSSREFS

Cf. A008288, which has g.f. 1/(1-x-x*y-x^2*y).

Cf. A078057 (row sums), A050146 (central terms).

Sequence in context: A135837 A027144 A158303 * A059370 A084534 A165899

Adjacent sequences:  A035604 A035605 A035606 * A035608 A035609 A035610

KEYWORD

nonn,easy,tabl

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from David W. Wilson

Maple program corrected and information added by Johannes W. Meijer, Aug 05 2011

STATUS

approved

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Last modified December 19 10:13 EST 2014. Contains 252186 sequences.