I was playing with counting meanders, and took a look at the paper http://www.math.uwaterloo.ca/~malacroi/Latex/Meanders.pdf In particular, I was interested in Table A.4, where M^(k)_n (1 <= k <= n) gives the number of closed meandric systems of order n (= 2n crossings) with k components. A008828 gives shows M in lexicographic order on (n, k).
From the development in the paper, we find
M^(1)_n = A005315(n) = number of closed meanders of order n. M^(n)_n = A000108(n) = Catalan(n) SUM(k = 1..n; M^(k)_n) = Catalan(n)^2 I found the following empirical relations between elements in the table, which I do not believe are in the paper: M^(n-1)_n = M^(n)_n * (2n(n-1))/(n+2) M^(n-2)_n = M^(n-1)_n * ((n-2)(n^2+7n-2))/((n+3)(n+4)) I strongly suspect these relations hold for all n, A008828 could be used to verify this for a few larger n. I would not know how to prove these. If these relations indeed hold, it suggests that there might be a similar relation between each element in the table and the one above it. If a few more of these relations could be worked out and some pattern found, we might be able to find a formula for A005315 (he said overoptimistically).