OFFSET
0,4
COMMENTS
Number of ways to distribute n pairs of parentheses into 2 distinct patterns where each pattern represents a Catalan ordering (A000108), and each pattern must contain at least one pair of parentheses.
If one of the groups is allowed to have no parentheses, we arrive at A000150 (with a different offset).
Analog of A216785 with Catalan number replacing connected graph counts.
From Petros Hadjicostas, Jul 27 2020: (Start)
It is proved in A050182 that A050182(n) = 1/(2*n + 4)*(binomial(2*n + 4, n) - [(n mod 2) == 0]*binomial(n + 2, n/2)).
Let C(x) = A(x) + 1 = Sum_{n >= 0} c(n)*x^n be the g.f. of the Catalan numbers A000108. Then C(x)^2 = (C(x) - 1)/x. Then (A(x) + 1)^2 = A(x)/x, and thus, A(x)^2 = -2*A(x) - 1 + A(x)/x. Thus, (A(x)^2 - A(x^2))/2 = (-2*A(x) - 1 + A(x)/x - A(x^2))/2.
Substituting A(x) = Sum_{n >= 1} c(n)*x^n in the above expression, we get (after some algebra) that a(n) = (-2*c(n) + c(n+1) - [(n mod 2) == 0]*c(n/2))/2 for n >= 1. It is then easy to prove that a(n) = 2*A050182(n-2) = (1/n)*(binomial(2*n, n-2) - [(n mod 2) == 0]*binomial(n, (n/2) - 1)) for n >= 2, thus proving the conjecture below. (End)
FORMULA
a(2*n+1) = A007223(2*n+1).
Conjecture: a(n) = 2*A050182(n-2) for n >= 2.
From Petros Hadjicostas, Jul 27 2020: (Start)
a(n) = (-2*c(n) + c(n+1) - [(n mod 2) == 0]*c(n/2))/2 for n >= 1, where c = A000108.
a(n) = (1/n)*(binomial(2*n, n-2) - [(n mod 2) == 0]*binomial(n, (n/2) - 1)) for n >= 2. (End)
MATHEMATICA
A[x_] = (1 - Sqrt[1 - 4x])/(2x) - 1;
CoefficientList[(A[x]^2 - A[x^2])/2 + O[x]^30, x] (* Jean-François Alcover, Apr 30 2023 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
R. J. Mathar, Jul 19 2016
STATUS
approved