cp's OEIS Frontend

From Petros Hadjicostas, Jan 14 2018: (Start)

For this sequence, if (b(n): n>=1) = BIK((a(n): n>=1)), then b(n) = a(n+1) for n>=1.

Let A(x) = Sum_{n>=1} a(n)*x^n be the g.f. for this sequence. For an explanation on how to derive the formula BIK(A(x)) = (1/2)*(A(x)/(1-A(x)) + (A(x) + A(x^2))/(1 - A(x^2))) from Bower's formulae in the link below about transforms, see the comments for sequence A001224. (For that sequence, the roles of sequences (a(n): n>=1) and (b(n): n>=1) are reversed.)

According to Bower's theory in the link below, we have boxes of different sizes and colors. The size of a box is determined by the number of balls it can hold. Two boxes of the same size and color are considered identical (indistinct and unlabeled). We place the boxes on a line that can be read in either direction; i.e., we have a reversible line.

Here, a(n) = number of colors a box holding n balls can be, while b(n) = number of ways of placing boxes in a line that can be read in either direction when the total number of balls is n.

According to C. G. Bower in the weblink below, a "[d]yslexic planar tree is a planar tree where each sub-rooted tree extending from a node can be read from left to right or right to left." A dyslexic planar tree "can be thought of as viewed by an observer who does not know left from right or as sub-rooted trees that can be turned around independent of the rest of the tree."

Given the above definition, a "ball" is a "node", a "box" is a "sub-rooted tree" (without the single root of the whole planar tree), and a "set of colors" for a "box" with n "balls" (= "nodes") is "the set of all dyslexic trees with n nodes". Hence, a(n) = number of all dyslexic planar trees with n nodes. On the other hand, b(n) = number of "reversible" arrangements of having sub-rooted dyslexic planar trees (= boxes on a reversible line) with a total of n-1 nodes in all subtrees (the n-th node is the single root of the whole tree). This means that b(n-1) = a(n) for n >= 2.

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Same as A001224 except that there a(2)=2 not 1. - N. J. A. Sloane, Mar 30 2015

This is an interleaving of A005207 and A051450. Thus a(2*m) = A005207(m) = (F(2*m-1) + F(m+1)) / 2, a(2*m - 1) = A051450(m) = (F(2*m) + F(m)) / 2 where F() are Fibonacci numbers (A000045). - Max Alekseyev, Jun 28 2006

The Kn11(n) and Kn21(n) sums, see A180662 for their definitions, of Losanitsch's triangle A034851 equal a(n), while the Kn12(n) and Kn22(n) sums equal (a(n+2)-A000012(n)) and the Kn13(n) and Kn23(n) sums equal (a(n+4)-A008619(n+4)). - Johannes W. Meijer, Jul 14 2011

a(n) is the number of homeomorphically irreducible caterpillars with n + 3 edges. - Christian Barrientos, Sep 12 2020

Apparently, beginning with a(3), number of non-equivalent 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)/(1-A(x)) + (A(x) + A(x^2))/(1-A(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 A-clusters 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/(1-B(x,y)) + (1 + B(x,y))*B(x^2, y^2)/(1-B(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/(1-B(1,y)) + (1 + B(1,y))*B(1,y^2)/(1-B(1,y^2))), which is equivalent to 2*B(1,y) = y + (1/2)*(B(1,y)/(1-B(1,y)) + (B(1,y) + B(1,y^2))/(1-B(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 Wedderburn-Etherington numbers, and apparently b(m=2, n) = A024206(n-1) for n>=2 (conjecture).

In Section 6 of their paper, Schöbel and Veselov (2014, 2015) prove that b(m,n) is the "number of non-equivalent faces of [the Stasheff polytope] K_n of codimension m-1." Apparently then, for n>=2 and k>=0, b(n-k,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 <= n-1 (see pp. 1269-1270 in Shobel and Veselov (2015)). Hence, for n>=2, Sum_{k=0..n-1} b(n-k, 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 [non-equivalent] canonical forms for separation coordinates on [hypersphere] S^n", which is the comment made by T. Copeland above.

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For an explanation on the meaning of clusters of types A, B, and C see Section 3 (pp. 78-81) in Devadoos and Read (2001). See also the comments for sequence A232206. - Petros Hadjicostas, Mar 02 2018

The only known value where a(n) is strictly less than the theoretical upper bound A001224(n+1) is a(6) = 11. - Andrew Howroyd, Dec 24 2019

From Petros Hadjicostas, Jan 14 2018: (Start)

For this sequence, if (b(n): n>=1) = BIK((a(n): n>=1)), then b(n) = a(n+1) for n>=1.

Let A(x) = Sum_{n>=1} a(n)*x^n be the g.f. for this sequence. For an explanation on how to derive the formula BIK(A(x)) = (1/2)*(A(x)/(1-A(x)) + (A(x) + A(x^2))/(1 - A(x^2))) from Bower's formulae in the link below about transforms, see the comments for sequence A001224. (For that sequence, the roles of sequences (a(n): n>=1) and (b(n): n>=1) are reversed.)

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From Petros Hadjicostas, Jan 14 2018: (Start)

For this sequence, if (b(n): n>=1) = BIK((a(n): n>=1)), then b(n) = a(n+2) for n>=1.

Let A(x) = Sum_{n>=1} a(n)*x^n be the g.f. for this sequence. For an explanation on how to derive the formula BIK(A(x)) = (1/2)*(A(x)/(1-A(x)) + (A(x) + A(x^2))/(1 - A(x^2))) from Bower's formulae in the link below about transforms, see the comments for sequence A001224. (For that sequence, the roles of sequences (a(n): n>=1) and (b(n): n>=1) are reversed.)

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