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.
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
Offset: 0
Examples
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
Links
- G. C. Greubel, Antidiagonals n = 0..50, 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.
- Armin Straub, Multivariate Apéry numbers and supercongruences of rational functions, Algebra & Number Theory, Vol. 8, No. 8 (2014), pp. 1985-2008; arXiv preprint, arXiv:1401.0854 [math.NT], 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).
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.)
Programs
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Magma
A:= func< n,k | (&+[(Binomial(n,j)*Binomial(n+k-j,k-j))^2: j in [0..n]]) >; // Array A143007:= func< n,k | A(n-k,k) >; // Antidiagonal triangle [A143007(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 05 2023
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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;
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Mathematica
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 *) -
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("")) -
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("")) -
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("")) -
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("")) -
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("")) /* End */ -
SageMath
def A(n,k): return sum((binomial(n,j)*binomial(n+k-j,k-j))^2 for j in range(n+1)) # array def A143007(n,k): return A(n-k,k) # antidiagonal triangle flatten([[A143007(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 05 2023
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} 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} 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)).
Sum_{k=0..n} T(n-k,k) = A227845(n) (antidiagonal sums). - Paul D. Hanna, Aug 27 2014
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).
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)
From Peter Bala, Jun 23 2023: (Start)
T(n,k) = Sum_{j = 0..n} C(n,j)^2 * C(n+k-j,k-j)^2.
T(n,k) = binomial(n+k,k)^2 * hypergeom([-n, -n, -k, -k],[-n - k, -n - k, 1], 1).
T(n,k) = hypergeom([n+1, -n, k+1, -k], [1, 1, 1], 1). (End)
From Peter Bala, Jun 28 2023: (Start)
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 ).
T(n,k) = A(n, k, n, k) in the notation of Straub, equation 7.
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.
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)
From G. C. Greubel, Oct 05 2023: (Start)
Let t(n, k) = T(n-k, k) be the antidiagonal triangle, then:
t(n, k) = t(n, n-k).
Sum_{k=0..floor(n/2)} t(n-k,k) = A246563(n).
t(2*n+1, n+1) = A352653(n+1). (End)
From Peter Bala, Sep 27 2024: (Start)
Let L denote the lower triangular array (l(n,k))n,k >= 0, where l(n, k) = (-1)^(n+k) * binomial(n, k)*binomial(n+k, k). (L is a signed version of A063007 and L = A063007 * A007318 ^(-1).)
Then the square array = L * transpose(A108625).
Extensions
Spelling/notation corrections by Charles R Greathouse IV, Mar 18 2010
Comments