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A143007 Square array, read by antidiagonals, where row n equals the crystal ball sequence for the 2*n-dimensional lattice A_n x A_n. 52
1, 1, 1, 1, 5, 1, 1, 13, 13, 1, 1, 25, 73, 25, 1, 1, 41, 253, 253, 41, 1, 1, 61, 661, 1445, 661, 61, 1, 1, 85, 1441, 5741, 5741, 1441, 85, 1, 1, 113, 2773, 17861, 33001, 17861, 2773, 113, 1, 1, 145, 4873, 46705, 142001, 142001, 46705, 4873, 145, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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 generalize 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.

LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux 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.

FORMULA

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 factorization 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.

From Paul D. Hanna, Aug 27 2014: (Start)

G.f. A(x,y) = Sum_{n>=0, k=0..n} T(n,k)*x^n*y^k can be expressed by:

(1) Sum_{n>=0} x^n * y^n / (1-x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * x^k]^2,

(2) Sum_{n>=0} x^n / (1 - x*y)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * x^k * y^k]^2,

(3) Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * y^k * Sum_{j=0..k} C(k,j)^2 * x^j,

(4) Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * y^(n-k) * Sum_{j=0..k} C(k,j)^2 * x^j * y^j.

(End)

EXAMPLE

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

MAPLE

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;

MATHEMATICA

t[n_, k_] := HypergeometricPFQ[ {-k, k+1, -n, n+1}, {1, 1, 1}, 1]; Table[ t[n-k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 06 2013 *)

PROG

(PARI) /* Print as a square array: */

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

for(n=0, 10, for(k=0, 10, print1(T(n, k), ", ")); print("")) \\ Paul D. Hanna, Aug 27 2014

(PARI) /* (1) G.f. A(x, y) when read as a triangle: */

{T(n, k)=local(A=1+x); A=sum(m=0, n, x^m * y^m / (1-x +x*O(x^n))^(2*m+1) * sum(k=0, m, binomial(m, k)^2*x^k)^2 ); polcoeff(polcoeff(A, n, x), k, y)}

for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print("")) \\ Paul D. Hanna, Aug 27 2014

(PARI) /* (2) G.f. A(x, y) when read as a triangle: */

{T(n, k)=local(A=1+x); A=sum(m=0, n, x^m/(1-x*y +x*O(x^n))^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * x^k * y^k)^2 ); polcoeff(polcoeff(A, n, x), k, y)}

for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print("")) \\ Paul D. Hanna, Aug 27 2014

(PARI) /* (3) G.f. A(x, y) when read as a triangle: */

{T(n, k)=local(A=1+x); A=sum(m=0, n, x^m*sum(k=0, m, binomial(m , k)^2 * y^k * sum(j=0, k, binomial(k, j)^2 * x^j)+x*O(x^n))); polcoeff(polcoeff(A, n, x), k, y)}

for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print("")) \\ Paul D. Hanna, Aug 27 2014

(PARI) /* (4) G.f. A(x, y) when read as a triangle: */

{T(n, k)=local(A=1+x); A=sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * y^(m-k) * sum(j=0, k, binomial(k, j)^2 * x^j * y^j)+x*O(x^n))); polcoeff(polcoeff(A, n, x), k, y)}

for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print("")) \\ Paul D. Hanna, Aug 27 2014

CROSSREFS

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).

Cf. A227845 (antidiagonal sums), A246464.

The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692,A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)

Sequence in context: A300035 A130227 A114123 * A152654 A176487 A272644

Adjacent sequences:  A143004 A143005 A143006 * A143008 A143009 A143010

KEYWORD

easy,nonn,tabl

AUTHOR

Peter Bala, Jul 22 2008

EXTENSIONS

Spelling/notation corrections by Charles R Greathouse IV, Mar 18 2010

STATUS

approved

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Last modified October 22 04:04 EDT 2018. Contains 316431 sequences. (Running on oeis4.)