A298637 Triangular array of a Catalan number variety: T(n,k) is the number of words consisting of n parentheses containing k well-balanced pairs.

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, 13, 121, 486, 1078, 1375, 891, 132, 14, 144, 650, 1664, 2574, 2288, 858, 15, 169, 847, 2457, 4459, 5005, 3003, 429

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.

Marko Riedel, Maple code for A298637 including enumeration, generating function, and two closed forms.

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

Row sums give A000079.

T(2n,n) gives A000108.

T(2n+1,n) gives A068875. T(n,1) gives A000290. T(2n,2) gives A280089.

Cf. A007318, A061554.

Sequence in context: A354265 A324336 A324752 * A034867 A323893 A329721

Adjacent sequences: A298634 A298635 A298636 * A298638 A298639 A298640

Keyword

nonn,tabf

Author

Marko Riedel, Jan 23 2018

Status

approved