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%I #63 Oct 05 2023 08:36:18
%S 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,
%T 661,61,1,1,85,1441,5741,5741,1441,85,1,1,113,2773,17861,33001,17861,
%U 2773,113,1,1,145,4873,46705,142001,142001,46705,4873,145,1
%N Square array, read by antidiagonals, where row n equals the crystal ball sequence for the 2*n-dimensional lattice A_n x A_n.
%C 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.
%C 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} 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} 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} (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)).
%C 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.
%H G. C. Greubel, <a href="/A143007/b143007.txt">Antidiagonals n = 0..50, flattened</a>
%H R. Bacher, P. de la Harpe and B. Venkov, <a href="http://archive.numdam.org/ARCHIVE/AIF/AIF_1999__49_3/AIF_1999__49_3_727_0/AIF_1999__49_3_727_0.pdf">Séries de croissance et séries d'Ehrhart associées aux réseaux de racines</a>, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
%H J. H. Conway and N. J. A. Sloane, <a href="http://citeseer.ist.psu.edu/75220.html">Low dimensional lattices VII Coordination sequences</a>, Proc. R. Soc. Lond., Ser. A, 453 (1997), 2369-2389.
%H Armin Straub, <a href="http://dx.doi.org/10.2140/ant.2014.8.1985">Multivariate Apéry numbers and supercongruences of rational functions</a>, Algebra & Number Theory, Vol. 8, No. 8 (2014), pp. 1985-2008; <a href="https://arxiv.org/abs/1401.0854">arXiv preprint</a>, arXiv:1401.0854 [math.NT], 2014.
%F T(n,k) = Sum_{j = 0..n} C(n+j,2*j)*C(2*j,j)^2*C(k+j,2*j).
%F The array is symmetric T(n,k) = T(k,n).
%F The main diagonal [1,5,73,1445,...] is the sequence of Apery numbers A005259.
%F 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).
%F 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].
%F Series acceleration formulas for zeta(3): Row n: zeta(3) = (1 + 1/2^3 + ... + 1/n^3) + Sum_{k >= 1} 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.
%F Main diagonal: zeta(3) = 6 * Sum_{n >= 1} 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} (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)).
%F Sum_{k=0..n} T(n-k,k) = A227845(n) (antidiagonal sums). - _Paul D. Hanna_, Aug 27 2014
%F The main superdiagonal numbers S(n) := T(n,n+1) appear to satisfy the supercongruences S(m*p^r - 1) == S(m*p^(r-1) - 1) (mod p^(3*r)) for prime p >= 5 and m, r in N (this is true: see A352653. - _Peter Bala_, Apr 16 2022).
%F From _Paul D. Hanna_, Aug 27 2014: (Start)
%F G.f. A(x,y) = Sum_{n>=0, k=0..n} T(n,k)*x^n*y^k can be expressed by:
%F (1) Sum_{n>=0} x^n * y^n / (1-x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * x^k]^2,
%F (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,
%F (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,
%F (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)
%F From _Peter Bala_, Jun 23 2023: (Start)
%F T(n,k) = Sum_{j = 0..n} C(n,j)^2 * C(n+k-j,k-j)^2.
%F T(n,k) = binomial(n+k,k)^2 * hypergeom([-n, -n, -k, -k],[-n - k, -n - k, 1], 1).
%F T(n,k) = hypergeom([n+1, -n, k+1, -k], [1, 1, 1], 1). (End)
%F From _Peter Bala_, Jun 28 2023: (Start)
%F T(n,k) = the coefficient of (x*z)^n*(y*t)^k in the expansion of 1/( (1 - x - y)*(1 - z - t) - x*y*z*t ).
%F T(n,k) = A(n, k, n, k) in the notation of Straub, equation 7.
%F The supercongruences T(n*p^r, k*p^r) == T(n*p^(r-1), k*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and positive integers n and k.
%F The formula T(n,k) = hypergeom([n+1, -n, k+1, -k], [1, 1, 1], 1) allows the table indexing to be extended to negative values of n and k; we have T(-n,k) = T(n-1,k) and T(n,-k) = T(n,k-1) leading to T(-n,-k) = T(n-1, k-1). (End)
%F From _G. C. Greubel_, Oct 05 2023: (Start)
%F Let t(n, k) = T(n-k, k) be the antidiagonal triangle, then:
%F t(n, k) = t(n, n-k).
%F Sum_{k=0..floor(n/2)} t(n-k,k) = A246563(n).
%F t(2*n+1, n+1) = A352653(n+1). (End)
%e The table begins
%e n\k|0...1.....2......3.......4.......5
%e ======================================
%e 0..|1...1.....1......1.......1.......1
%e 1..|1...5....13.....25......41......61 A001844
%e 2..|1..13....73....253.....661....1441 A143008
%e 3..|1..25...253...1445....5741...17861 A143009
%e 4..|1..41...661...5741...33001..142001 A143010
%e 5..|1..61..1441..17861..142001..819005 A143011
%e ........
%e Example row 1 [1,5,13,...]:
%e 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, ... .
%e . . . . . . . . . . .
%e . . . . . . . . . . .
%e . . . . . 2 . . . . .
%e . . . . 2 1 2 . . . .
%e . . . 2 1 0 1 2 . . .
%e . . . . 2 1 2 . . . .
%e . . . . . 2 . . . . .
%e . . . . . . . . . . .
%e . . . . . . . . . . .
%e 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).
%e Read as a triangle the array begins
%e n\k|0...1....2....3...4...5
%e ===========================
%e 0..|1
%e 1..|1...1
%e 2..|1...5....1
%e 3..|1..13...13....1
%e 4..|1..25...73...25...1
%e 5..|1..41..253..253..41...1
%p 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;
%t T[n_, k_]:= HypergeometricPFQ[{-k, k+1, -n, n+1}, {1, 1, 1}, 1]; Table[T[n-k, k], {n,0,12}, {k,0,n}]//Flatten (* _Jean-François Alcover_, Mar 06 2013 *)
%o /* Programs from _Paul D. Hanna_, Aug 27 2014 */
%o (PARI) /* Print as a square array: */
%o {T(n, k)=sum(j=0, n, binomial(n+j, 2*j)*binomial(2*j, j)^2*binomial(k+j, 2*j))}
%o for(n=0, 10, for(k=0,10, print1(T(n,k), ", "));print(""))
%o (PARI) /* (1) G.f. A(x,y) when read as a triangle: */
%o {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)}
%o for(n=0, 10, for(k=0,n, print1(T(n,k), ", "));print(""))
%o (PARI) /* (2) G.f. A(x,y) when read as a triangle: */
%o {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)}
%o for(n=0, 10, for(k=0,n, print1(T(n,k), ", "));print(""))
%o (PARI) /* (3) G.f. A(x,y) when read as a triangle: */
%o {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)}
%o for(n=0, 10, for(k=0,n, print1(T(n,k), ", "));print(""))
%o (PARI) /* (4) G.f. A(x,y) when read as a triangle: */
%o {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)}
%o for(n=0, 10, for(k=0,n, print1(T(n,k), ", "));print(""))
%o /* End */
%o (Magma)
%o A:= func< n,k | (&+[(Binomial(n,j)*Binomial(n+k-j,k-j))^2: j in [0..n]]) >; // Array
%o A143007:= func< n,k | A(n-k,k) >; // Anti-diagonal triangle
%o [A143007(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Oct 05 2023
%o (SageMath)
%o def A(n,k): return sum((binomial(n,j)*binomial(n+k-j,k-j))^2 for j in range(n+1)) # array
%o def A143007(n,k): return A(n-k,k) # anti-diagonal triangle
%o flatten([[A143007(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Oct 05 2023
%Y 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).
%Y Cf. A227845 (antidiagonal sums), A246464.
%Y 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.)
%Y Cf. A246563, A352653.
%K easy,nonn,tabl
%O 0,5
%A _Peter Bala_, Jul 22 2008
%E Spelling/notation corrections by _Charles R Greathouse IV_, Mar 18 2010
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