
COMMENTS

Apparently, beginning with a(3), number of nonequivalent canonical forms of separation coordinates on the hyperspheres. Cf. Schöbel and Veselov for this and other interpretations.  Tom Copeland, Nov 21 2017
From Petros Hadjicostas, Jan 17 2018: (Start)
Let A(x) = Sum_{n>=1} a(n)*x^n. For a derivation of the formula BIK(A(x)) = (1/2)*(A(x)/(1A(x)) + (A(x) + A(x^2))/(1A(x^2))), see the comments for sequence A001224 and the weblink below containing Bower's theory of transforms.
We clarify the comment by T. Copeland above. Consider the material in Section 3 of Devadoss and Read (2001). According to their terminology, let b(m,n) be "the number of Aclusters having m cells and n outside edges not counting the root edge." Let B(x,y) = Sum_{m>=0, n>=0} b(m,n)*x^m*y^n. (See p. 78 in their paper, where they use the notations a_{m,n} and A(x,y) rather than b(m,n) and B(x,y), respectively, that we use here.)
On p. 79 (Eq. (3.1)) of their paper, they prove that B(x,y) = y + (x/2)*(B(x,y)^2/(1B(x,y)) + (1 + B(x,y))*B(x^2, y^2)/(1B(x^2,y^2))). Unfortunately, the factor x in the previous formula is left out (i.e., it is a typo), and the same typo is reproduced in Schöbel and Veselov (2014, 2015).
Using Table 2 (p. 92) from Devadoss and Read (2001) (and the material on p. 79), we get that B(x,y) = y+ x*y^2 + (x^2 + x)*y^3 + (2*x^3 + 3*x^2 + x)*y^4 + (3*x^4 + 8*x^3 + 5*x^2 + x)*y^5 + ...
We claim that a(n) = Sum_{m>=0} b(m,n) and A(y) = Sum_{n>=1} a(n)*y^n = B(x=1, y). To prove these claims, note that, for x=1, the above series becomes B(x=1,y) = y + y^2 + 2*y^3 + 6*y^4 + 17*y^5 + ..., while the functional equation above becomes B(1, y) = y + (1/2)*(B(1,y)^2/(1B(1,y)) + (1 + B(1,y))*B(1,y^2)/(1B(1,y^2))), which is equivalent to 2*B(1,y) = y + (1/2)*(B(1,y)/(1B(1,y)) + (B(1,y) + B(1,y^2))/(1B(1,y^2))). The latter formula is the one given in the formula section below (derived from Bower's theory) with x replaced with y and A(x) replaced with B(1,y). This proves that B(x=1, y) = A(y), from which we can easily get that a(n) = Sum_{m>=0} b(m,n).
Note that b(m=0, n) = 0 for n <> 1, but b(m=0, n=1) = 1; b(m,n) = 0 when m >= n >= 1; and b(m=1, n) = 1 for n>=2. Also, b(m,m+1) = A001190(m+1) for m>=1, which are the WedderburnEtherington numbers, and apparently b(m=2, n) = A024206(n1) for n>=2 (conjecture).
In Section 6 of their paper, Schöbel and Veselov (2014, 2015) prove that b(m,n) is the "number of nonequivalent faces of [the Stasheff polytope] K_n of codimension m1." Apparently then, for n>=2 and k>=0, b(nk,n+1) is the "number of canonical forms for separation coordinates of [hypersphere] S^n" with k "independent continuous parameters". For k=0 and n>=2, b(n,n+1) = A001190(n+1) = "number of canonical forms for separation coordinates" of hypersphere S^n with 0 continuous parameters.
It turns out that for k, the number of continuous parameters of S^n, we have 0 <= k <= n1 (see pp. 12691270 in Shobel and Veselov (2015)). Hence, for n>=2, Sum_{k=0..n1} b(nk, n+1) = Sum_{m=1..n} b(m, n+1) = Sum_{m=0..n} b(m, n+1) = a(n+1) (see above). As a result, for n>=2, a(n+1) is the "total number of [nonequivalent] canonical forms for separation coordinates on [hypersphere] S^n", which is the comment made by T. Copeland above.
(End)
For an explanation on the meaning of clusters of types A, B, and C see Section 3 (pp. 7881) in Devadoos and Read (2001). See also the comments for sequence A232206.  Petros Hadjicostas, Mar 02 2018
