cp's OEIS Frontend

A142992 Square array, read by ascending antidiagonals, of the crystal ball sequences for the root lattices of type C_n.

%I A142992 #44 Jun 23 2024 06:37:13
%S A142992 1,1,1,1,3,1,1,9,5,1,1,19,25,7,1,1,33,85,49,9,1,1,51,225,231,81,11,1,
%T A142992 1,73,501,833,489,121,13,1,1,99,985,2471,2241,891,169,15,1,1,129,1765,
%U A142992 6321,8361,4961,1469,225,17,1
%N A142992 Square array, read by ascending antidiagonals, of the crystal ball sequences for the root lattices of type C_n.
%C A142992 The lattice C_n consists of all integer lattice points v = (x_1,...,x_n) in Z^n such that the sum x_1 + ... + x_n is even. Let ||v|| = 1/2 * Sum_{i = 1..n} |x_i|; this defines a norm on C_n. The k-th term of the crystal ball sequence of C_n gives the number of lattice points v in C_n with ||v|| <= k [Bacher et al.]. The case n = 2 is illustrated in the Example section below.
%C A142992 This array has a remarkable relationship with the constant log(2). The row, column and (conjecturally) the diagonal entries of the array occur in series acceleration formulas for log(2) (see the Formula section below for some examples).
%C A142992 See A103884 for the table of coordination sequences of the C_n lattices. For the crystal ball sequences for the A_n and D_n lattices see A108625 and A108553 respectively. For the crystal ball sequences for the product lattices A_1 x ... x A_1(n copies) and A_n x A_n see A008288 and A143007 respectively.
%H A142992 Seiichi Manyama, <a href="/A142992/b142992.txt">Rows n = 0..139, flattened</a>
%H A142992 R. Bacher, P. de la Harpe and B. Venkov, <a href="https://doi.org/10.5802/aif.1689">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 A142992 D. Bump, K. Choi, P. Kurlberg and J. Vaaler, <a href="https://www.researchgate.net/publication/225542192_A_local_Riemann_hypothesis_I">A local Riemann hypothesis, I</a>, Math. Zeit. 233, (2000), 1-19.
%H A142992 J. H. Conway and N. J. A. Sloane, <a href="http://neilsloane.com/doc/Me220.pdf">Low dimensional lattices VII Coordination sequences</a>, Proc. R. Soc. Lond., Ser. A, 453 (1997), 2369-2389.
%F A142992 T(n,k) = Sum_{i = 0..n} C(2*n,2*i)*C(k+i,n).
%F A142992 O.g.f. for row n: 1/(1-x)^(n+1) * Sum_{k = 0..n} C(2*n,2*k)*x^k = 1/(1-x) * T(n,(1+x)/(1-x)), where T(n,x) denotes the Chebyshev polynomial of the first kind.
%F A142992 O.g.f. for the array: 1/(1-x) * {(1-t) - x*(1+t)}/{(1-t)^2 - x*(1+t)^2} = (1+x+x^2+x^3+...) + (1+3*x+5*x^2+7*x^3+...)*t + (1+9*x+25*x^2+49*x^3+...)*t^2 + ... .
%F A142992 Row n of the array has the form [p_n(0),p_n(1),p_n(2),...], where the polynomial function p_n(x) = Sum_{k = 0..n} C(2*n,2*k)*C(x+k,n). The first few are p_0(x) = 1, p_1(x) = 2*x+1, p_2(x) = (2*x+1)^2, p_3(x) = (2*x+1)*(8*x^2+8*x+3)/3 and p_4(x) = (2*x+1)^2*(4*x^2+4*x+3)/3.
%F A142992 Alternative expressions for p_n(x) include p_n(x) = Sum_{k = 0..n} 2^(2*k)*n/(n+k)*C(n+k,2*k)*C(x,k) and p_n(x) = Sum_{k = 1..n} 2^(k-1)*C(n-1,k-1)*C(2*x+1,k).
%F A142992 The polynomials p_n(x) satisfy the 3-term recurrence relation n*p_n(x) = 2*(2*x+1)*p_(n-1)(x)+(n-2)*p(n-2)(x) for n >= 2; their generating function is 1/2*((1+t)/(1-t))^(2*x+1) = 1/2 + (2*x+1)*t + (2*x+1)^2*t^2 + (2*x+1)*(8*x^2+8*x+3)/3*t^3 + ... . Thus p_n(x) is, apart from a constant factor, the Meixner polynomial of the first kind M_n(2*x+1;b,c) at b = 0, c = -1. Compare with A142979.
%F A142992 The polynomial p_n(x) is the unique polynomial solution to the difference equation (2*x+1)*{f(x+1/2) - f(x-1/2)} = 2*n*f(x), normalized so that f(0) = 1. The function p_n(x) is also the unique polynomial solution to the difference equation (2*x+1)*{(x+1)*f(x+1) + x*f(x-1)} = ((2*x+1)^2 + 2*n^2)*f(x), normalized so that f(0) = 1.
%F A142992 The zeros of p_n(x) lie on the vertical line Re x = -1/2 in the complex plane, that is, the polynomials p_n(x-1), n = 1,2,3,..., satisfy a Riemann hypothesis (adapt the proof of the lemma on p.4 of [BUMP et al.]).
%F A142992 For n > 0, the entries in row n of the array occur in series acceleration formulas for log(2): 2*log(2) = 1 + (1/2 - 1/6 +...+(-1)^n/(n*(n-1))) + (-1)^(n+1)*Sum_{k >= 1} 1/(k*T(n,k-1)*T(n,k)). For example, the fourth row of the table (n = 3) gives 2*log(2) = 4/3 + 1/(1*1*19) + 1/(2*19*85) + 1/(3*85*231) + ... .
%F A142992 The corresponding result for column k is 2*log(2) = 1 + (1/(1*3) + 1/(2*3*5) +...+ 1/(k*(2*k-1)*(2k+1)) + (2*k+1)*Sum_{n >= 1} (-1)^(n+1)/(n*(n+1)*T(n,k)* T(n+1,k)).
%F A142992 For example, the third column of the table (k = 2) gives 2*log(2) = 41/30 + 5*(1/(1*2*5*25) - 1/(2*3*25*85) + 1/(3*4*85*225) - ... ).
%F A142992 For the main diagonal calculation suggests the result: 2*log(2) = 4/3 + Sum_{n >= 1} (-1)^(n+1)*(5*n+3)/(n*(n+1)*T(n,n)*T(n+1,n+1)).
%F A142992 Similar series acceleration formulas for log(2) come from the row, column and diagonal entries of the square array of Delannoy numbers, A008288 (which may viewed as the array of crystal ball sequences for the product lattices A_1 x...x A_1). For corresponding results for the constants zeta(2) and zeta(3) see A108625 and A143007 respectively.
%e A142992 The square array begins
%e A142992 n\k|0...1....2.....3.....4......5
%e A142992 =================================
%e A142992 .0.|1...1....1.....1.....1......1
%e A142992 .1.|1...3....5.....7.....9.....11
%e A142992 .2.|1...9...25....49....81....121 A016754
%e A142992 .3.|1..19...85...231...489....891 A063496
%e A142992 .4.|1..33..225...833..2241...4961 A142993
%e A142992 .5.|1..51..501..2471..8361..22363 A142994
%e A142992 ...
%e A142992 Triangular array begins
%e A142992 n\k|0...1...2...3...4...5
%e A142992 =========================
%e A142992 .0.|1
%e A142992 .1.|1...1
%e A142992 .2.|1...3...1
%e A142992 .3.|1...9...5...1
%e A142992 .4.|1..19..25...7...1
%e A142992 .5.|1..33..85..49...9...1
%e A142992 Case n = 2: The C_2 lattice consists of all integer lattice points v = (x,y) in Z x Z such that x + y is even, equipped with the taxicab type norm ||v|| = 1/2 * (|x| + |y|). There are 8 lattice points (marked with a 1 on the figure below) satisfying ||v|| = 1 and 16 lattice points (marked with a 2 on the figure) satisfying ||v|| = 2. Hence the crystal ball sequence for the C_2 lattice (row 2 of the table) begins 1, 1+8 = 9, 1+8+16 = 25, ... .
%e A142992 . . . . . . . . . . .
%e A142992 . . . . . 2 . . . . .
%e A142992 . . . . 2 . 2 . . . .
%e A142992 . . . 2 . 1 . 2 . . .
%e A142992 . . 2 . 1 . 1 . 2 . .
%e A142992 . 2 . 1 . 0 . 1 . 2 .
%e A142992 . . 2 . 1 . 1 . 2 . .
%e A142992 . . . 2 . 1 . 2 . . .
%e A142992 . . . . 2 . 2 . . . .
%e A142992 . . . . . 2 . . . . .
%e A142992 . . . . . . . . . . .
%p A142992 with combinat: T := (n,k) -> add(binomial(2n,2i)*binomial(k+i,n),i = 0..n): for n from 0 to 9 do seq(T(n,k), k = 0..9) end do;
%t A142992 t[n_, k_] := Sum[ Binomial[2*n, 2*i]*Binomial[k+i, n], {i, 0, n}]; Table[t[n-k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Mar 06 2013 *)
%Y A142992 Row 2-10 give A016754, A063496, A142993, A142994, A305549, A305721, A305722, A305723, A305724.
%Y A142992 Cf. A008288, A103884, A108625, A142979, A143007.
%K A142992 easy,nonn,tabl
%O A142992 0,5
%A A142992 _Peter Bala_, Jul 18 2008