Parentheses

https://cs.stackexchange.com/questions/77250/language-of-cfg-s-to-as-asbs-varepsilon?rq=1

Parenthesis
found here https://en.wikipedia.org/wiki/Context-free_grammar#Well-formed_parentheses

1, 1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718,

https://oeis.org/A155069

ambiguous : $$ S= ab + aSb + SS $$

found here https://en.wikipedia.org/wiki/Dyck_language

1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012,

https://oeis.org/A000108

non-ambiguous : $$ S = 1 + aSbS $$

Seven Trees in One, 1+1+2+5+14+42+...
This is my note in Wikipedia, introducing a divergent series that has a complex "sum". https://en.wikipedia.org/wiki/Talk:1_%2B_2_%2B_4_%2B_8_%2B_%E2%8B%AF#1.2B1.2B2.2B5.2B14.2B42.2B...

The Euler heuristic for 1 + 2 + 4 + 8 + 16+... = -1 is
 * $$\begin{array}{rcl}

s & = &\displaystyle 1+2+4+8+\cdots \\[1em] & = &\displaystyle 1+2\cdot(1+2+4+8+\cdots) \\[1em] & = &\displaystyle 1+2s \end{array}$$

There is an interpretation for this sum regarding binary words. A binary word on{a,b} either is null or it starts with a or it starts with b : S = 1 + S + S

The sum of Catalan numbers has a similar heuristic, following somehow operations with binary trees :


 * $$\begin{array}{rcl}

C     & = &\displaystyle 0+1+1+2+5+14+42+\cdots \\[1em] C*C   & = &\displaystyle 0+0+1+2+5+14+42+\cdots \\[1em] 1     & = &\displaystyle 0+1+0+0+0+0+0+\cdots \\[1em] C*C+1 & = &\displaystyle 0+1+1+2+5+14+42+\cdots = C \\[1em] \end{array}$$

so the sum is

$$C = \frac{1 \pm i\sqrt{3}} {2}$$

the nice part is that after further manipulations (product, sum, left-right shifting) one can get :
 * $$ C^7 = C $$

I have translated in terms of divergent series the material here : http://arxiv.org/pdf/math/0212377v1.pdf and here

http://arxiv.org/pdf/math/9405205v1.pdf that is not about divergent series. Nevertheless, manipulating binary trees reflects in manipulating the divergent series above.

Here is my question : does this make some sense to you ? Here, unlike other real "sums", the complex "sum" of a series is useful to further calculus. Is this the next level of heuristics of divergent series ? thanks :)Nboykou (talk) 00:13, 8 July 2015 (UTC)

Catalan Triangle
A009766 Catalan's triangle T(n,k) (read by rows): each term is the sum of the entries above and to the left, i.e., T(n,k) = Sum_{j=0..k} T(n-1,j).

$${n-k+1 \over n+1 }{n+k \choose k}$$

$${n \choose k} = {n-1 \choose k} + {n \choose k-1}$$

Row sums as well as the sums of the squares of the shallow diagonals give Catalan numbers A000108 C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!). Also called Segner numbers.

1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796,

https://brilliant.org/wiki/catalan-numbers/