0,5

The A_n lattice consists of all vectors v = (x_1,...,x_(n+1)) in Z^(n+1) such that x_1 + ... + x_(n+1) = 0. The lattice is equipped with the norm ||v|| = 1/2*(|x_1| + ... + |x_(n+1)|). Pairs of lattice points (v,w) in the product lattice A_n x A_n have norm ||(v,w)|| = ||v|| + ||w||. Then the k_th term in the crystal ball sequence for the A_n x A_n lattice gives the number of such pairs (v,w) for which ||(v,w)|| is less than or equal to k.

This array has a remarkable relationship with Apery's constant zeta(3). The row (or column) and main diagonal entries of the array occur in series acceleration formulas for zeta(3). For row n entries there holds zeta(3) = (1+1/2^3+...+1/n^3) + sum {k = 1..inf} 1/(k^3*T(n,k-1)*T(n,k)). Also, as consequence of Apery's proof of the irrationality of zeta(3), we have a series acceleration formula along the main diagonal of the table: zeta(3) = 6 * sum {n = 1..inf} 1/(n^3*T(n-1,n-1)*T(n,n)). Apery's result appears to generalise to the other diagonals of the table. Calculation suggests the following result may hold: zeta(3) = 1+1/2^3 + ... +1/k^3 + sum {n = 1..inf} (2*n+k)*(3*n^2+3*n*k+k^2)/(n^3*(n+k)^3*T(n-1,n+k-1)*T(n,n+k)).

For the corresponding results for the constant zeta(2), related to the crystal ball sequences of the lattices A_n, see A108625. For corresponding results for log(2), coming from either the crystal ball sequences of the hypercubic lattices A_1 x ... x A_1 or the lattices of type C_n, see A008288 and A142992 respectively.

R. Bacher, P. de la Harpe and B. Venkov, Series de croissance et series d'Ehrhart associees aux reseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.

J. H. Conway and N. J. A. Sloane, Low dimensional lattices VII Coordination sequences, Proc. R. Soc. Lond., Ser. A, 453 (1997), 2369-2389.

T(n,k) = sum {j = 0..n} C(n+j,2*j)*C(2*j,j)^2*C(k+j,2*j).

The array is symmetric T(n,k) = T(k,n).

The main diagonal [1,5,73,1445,...] is the sequence of Apery numbers A005259.

The entries in the k_th column satisfy the Apery-like recursion n^3*T(n,k) + (n-1)^3*T(n-2,k) = (2*n-1)*(n^2-n+1+2*k^2+2*k)*T(n-1,k).

The LDU factorisation of the square array is L * D * transpose(L), where L is the lower triangular array A085478 and D is the diagonal matrix diag(C(2n,n)^2). O.g.f. for row n: The generating function for the coordination sequence of the lattice A_n is [sum {k = 0..n} C(n,k)^2*x^k]/(1-x)^n. Thus the generating function for the coordination sequence of the product lattice A_n x A_n is {[sum {k = 0..n} C(n,k)^2*x^k]/(1-x)^n}^2 and hence the generating function for row n of this array, the crystal ball sequence of the lattice A_n x A_n, equals [sum {k = 0..n} C(n,k)^2*x^k]^2/(1-x)^(2n+1) = 1/(1-x)*[Legendre_P(n,(1+x)/(1-x))]^2. See [Conway & Sloane].

Series acceleration formulas for zeta(3): Row n: zeta(3) = (1+1/2^3+...+1/n^3) + sum {k = 1..inf} 1/(k^3*T(n,k-1)*T(n,k)), n = 0,1,2,... . For example, the fourth row of the table (n = 3) gives zeta(3) = (1 + 1/2^3 + 1/3^3) + 1/(1^3*1*25) + 1/(2^3*25*253) + 1/(3^3*253*1445) + ... . See A143003 for further details.

Main diagonal: zeta(3) = 6 * sum {n = 1..inf} 1/(n^3*T(n-1,n-1)*T(n,n)). Conjectural result for other diagonals: zeta(3) = 1+1/2^3 + ... +1/k^3 + sum {n = 1..inf} (2*n+k)*(3*n^2+3*n*k+k^2)/(n^3*(n+k)^3*T(n-1,n+k-1)*T(n,n+k)).

The main superdiagonal numbers S(n):= T(n,n+1) appear to satisfy the super congruences S(m*p^r - 1) == S(m*p^(r-1) - 1) (mod p^(3*r)) for prime p >=5 and m,r in N.

The table begins

n\k|0...1.....2......3.......4.......5

======================================

0..|1...1.....1......1.......1.......1

1..|1...5....13.....25......41......61 A001844

2..|1..13....73....253.....661....1441 A143008

3..|1..25...253...1445....5741...17861 A143009

4..|1..41...661...5741...33001..142001 A143010

5..|1..61..1441..17861..142001..819005 A143011

........

Example row 1 [1,5,13,...]:

The lattice A_1 x A_1 is equivalent to the square lattice of all integer lattice points v = (x,y) in Z x Z equipped with the taxicab norm ||v|| = (|x| + |y|). There are 4 lattice points (marked with a 1 on the figure below) satisfying ||v|| = 1 and 8 lattice points (marked with a 2 on the figure) satisfying ||v|| = 2. Hence the crystal ball sequence for the A_1 x A_1 lattice begins 1, 1+4 = 5, 1+4+8 = 13, ... .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . 2 . . . . .

. . . . 2 1 2 . . . .

. . . 2 1 0 1 2 . . .

. . . . 2 1 2 . . . .

. . . . . 2 . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

Row 1 = [1,5,13,...] is the sequence of partial sums of A008574; row 2 = [1,13,73,...] is the sequence of partial sums of A008530, so row 2 is the crystal ball sequence for the lattice A_2 x A_2 (the 4-dimensional di-isohexagonal orthogonal lattice).

Read as a triangle the array begins

n\k|0...1....2....3...4...5

===========================

0..|1

1..|1...1

2..|1...5....1

3..|1..13...13....1

4..|1..25...73...25...1

5..|1..41..253..253..41...1

with(combinat): T:= (n, k) -> add(binomial(n+j, 2*j)*binomial(2*j, j)^2*binomial(k+j, 2*j), j = 0..n): for n from 0 to 9 do seq(T(n, k), k = 0..9) end do;

Cf. A001844 (row 1), A005259 (main diagonal), A008288, A008530 (first differences of row 2), A008574 (first differences of row 1), A085478, A108625, A142992, A143003, A143004, A143005, A143006, A143008 (row 2), A143009 (row 3), A142010 (row 4), A143011 (row 5).

Sequence in context: A146987 A130227 A114123 this_sequence A152654 A157177 A119725

Adjacent sequences: A143004 A143005 A143006 this_sequence A143008 A143009 A143010

easy,nonn,tabl

Peter Bala (pbala(AT)toucansurf.com), Jul 22 2008