OFFSET
0,4
COMMENTS
A well-formed bracketed word is a properly nested word using the symbol * together with four types of matching unary brackets; the condition "no empty bracket pair" means that every matching bracket pair encloses at least one symbol.
Under this interpretation there is no object of total length 0, hence the sequence is naturally indexed from n=1.
Equivalently, a(n) is the number of operator monomials of total length n generated from one indeterminate * by associative multiplication and four noncommuting unary operators P_1, P_2, P_3, P_4, where * has length 1 and each application of P_i contributes 2 to the total length.
Equivalently, for n>=1, a(n) counts peakless Motzkin paths of length n with 4-colored up steps; here "peakless" means that the path contains no occurrence of UD.
LINKS
Yu Hin Au and Murray R. Bremner, Enumerating Multi-Operator Monomials in Commutative and Noncommutative Settings, arXiv:2604.25731 [math.CO], 2026. See pp. 4, 9 (Table 2).
FORMULA
G.f.: 1 + A(x) satisfies A(x) = x + x*A(x) + 4*x^2*A(x) + 4*x^2*A(x)^2.
G.f.: 1 + (1 - x - 4*x^2 - sqrt((1 - x - 4*x^2)^2 - 16*x^3))/(8*x^2).
a(n) = Sum_{k=0..floor((n-1)/2)} 4^k*N(n-k,k), where N(m,k) = A001263(m,k) = binomial(m,k)*binomial(m,k+1)/m are the Narayana numbers.
a(0)=a(1)=1; for n>=2, a(n) = a(n-1) + 4*a(n-2) + 4*Sum_{i=1..n-3} a(i)*a(n-2-i), with a(m)=0 for m<=0.
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Yu Hin Au, Apr 08 2026
EXTENSIONS
a(0)=1 inserted by Sean A. Irvine, Apr 15 2026
STATUS
approved