
COMMENTS

R(n,k) is the number of ordered pairs (x,y) of integers x,y satisfying 1<=x<=k, 1<=y<=k, and x*y<=n.
Limiting row: A000618=(1,3,5,8,10,14,16,20,...).
Row 1: A000027
Row 2: A032766
Row 3: A106252
Row 4: A211703
Row 5: A211704
R(n,n)=A000618(n)
...
For n>=1, row n is a homogeneous linear recurrence sequence of order A005728(n), and it exemplifies a certain class, C, of recurrences which are palindromic (in the sense given below). The class depends on sequences s having nth term [(n^k)/j], where k and j are arbitrary fixed positive integers and [ ] = floor . The characteristic polynomial of s is (x^j1)(x1)^k, which is a palindromic polynomial (sometimes called a reciprocal polynomial). The class C consists of sequences u given by the form
...
u(n) = c(1)*[r(1)*n^k(1)] + ... + c(m)*[r(m)*n^k(m)],
...
where c(i) are integers and r(i) are rational numbers. Assume that r(i) is in lowest terms, and let j(i) be its denominator. Then the characteristic polynomial of u is the least common multiple of all the irreducible (over the integers) factors of all the polynomials (x^j(i)1)(x1)^k(i). As all such factors are palindromic (indeed, they are all cyclotomic polynomials), the characteristic polynomial of u is also palindromic. In other words, if the generating function of u is written as p(x)/q(x), then q(x) is a palindromic polynomial.
Thus, if q(x) = q(h)x^h + ... + q(1)x + q(0),
then (q(h), q(h1), ..., q(1), q(0)) is palindromic, and consequently, the recurrence coefficients for u, after excluding q(0); i.e., ( q(h1), ...  q(1)), are palindromic. For example, row 3 of A211701 has the following recurrence: u(n)=u(n2)+u(n3)u(n5), for which q(x)=x^5x^3x^2+1, with recurrence coefficients (0,1,1,0,1).
Recurrence coefficients (palindromic after excluding the last term) are shown here:
for row 1: (2, 1)
for row 2: (1 ,1, 1)
for row 3: (0, 1, 1, 0, 1)
for row 4: (0, 0, 1, 1, 0, 0, 1)
for row 5: (1, 1, 0, 1, 2, 2, 1, 0, 1, 1, 1)
for row 6: (0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1)
for row 7: (1, 2, 2, 2, 1, 0, 2, 3, 4, 4, 3, 2,
0, 1, 2, 2, 2, 1, 1)
for row 13: (2,4,7,12,18,27,37,50,64,80,95,
111,123,133,137,136,126,110,84,52,
12,32,80,127,173,213,246,269,281,281,269,
246,213,173,127,80,32,12,52,84,110,
126,136,137,133,123,111,95,80,64,
50,37,27,18,12,7,4,2,1)
