OFFSET
0,2
COMMENTS
A well-balanced run in a word of parentheses is a maximal run where every initial segment of the run has at least as many left parentheses as right ones and the number of open parentheses is the same as that of closed ones. The variable k in the sequence definition is the sum of the count of balanced pairs in all maximal runs in the word and n is the length of the word. Runs are maximal substrings counted by ordinary Catalan numbers.
LINKS
Alois P. Heinz, Rows n = 0..200, flattened
Toufik Mansour, Armend Sh. Shabani, Bargraphs in bargraphs, Turkish Journal of Mathematics (2018) Vol. 42, Issue 5, 2763-2773.
Marko Riedel et al., Generalisation for Catalan number.
FORMULA
T(n,k) = ((n+1-2*k)^2/(n+1))*C(n+1,k) where 0 <= k <= floor(n/2).
Bivariate o.g.f. is C(u*z^2)/(1-z*C(u*z^2))^2 with u counting pairs of parentheses and z counting total word length where C(z) = (1-sqrt(1-4*z))/(2*z) is the o.g.f. of the Catalan numbers.
T(2*k,k) = C(k), the k-th Catalan number.
T(n,0) = n+1 by construction.
EXAMPLE
The word ))))(()(()))((() contains five well-balanced pairs of parentheses.
Triangular array T(n,k) begins:
1;
2;
3, 1;
4, 4;
5, 9, 2;
6, 16, 10;
7, 25, 27, 5;
8, 36, 56, 28;
9, 49, 100, 84, 14;
10, 64, 162, 192, 84;
11, 81, 245, 375, 270, 42;
12, 100, 352, 660, 660, 264;
MAPLE
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
`if`(i>0, x, 1)*b(n-1, max(0, i-1))+b(n-1, i+1)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..16); # Alois P. Heinz, Jan 23 2018
MATHEMATICA
Table[((n + 1 - 2 k)^2/(n + 1)) Binomial[n + 1, k], {n, 0, 17}, {k, 0, Floor[n/2]}] // Flatten (* Michael De Vlieger, Jan 23 2018 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Marko Riedel, Jan 23 2018
STATUS
approved