A394941 - OEIS

%I #22 May 02 2026 10:10:06

%S 1,1,1,5,13,41,137,445,1525,5249,18321,64821,231069,831129,3010137,

%T 10968429,40189957,147969137,547163873,2031245413,7567313965,

%U 28282674185,106016940841,398474757149,1501425771797,5670242384737,21459558843697,81375882450581,309148795708605,1176476820260985

%N Number of well-formed bracketed words of total length n built from the symbol * (of length 1) and four unary bracket types, with no empty bracket pair.

%C 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.

%C Under this interpretation there is no object of total length 0, hence the sequence is naturally indexed from n=1.

%C 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.

%C 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.

%H Yu Hin Au and Murray R. Bremner, <a href="https://arxiv.org/abs/2604.25731">Enumerating Multi-Operator Monomials in Commutative and Noncommutative Settings</a>, arXiv:2604.25731 [math.CO], 2026. See pp. 4, 9 (Table 2).

%F G.f.: 1 + A(x) satisfies A(x) = x + x*A(x) + 4*x^2*A(x) + 4*x^2*A(x)^2.

%F G.f.: 1 + (1 - x - 4*x^2 - sqrt((1 - x - 4*x^2)^2 - 16*x^3))/(8*x^2).

%F 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.

%F 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.

%Y Cf. A001263, A004148, A394939, A394940, A059231.

%K nonn,easy

%O 0,4

%A _Yu Hin Au_, Apr 08 2026

%E a(0)=1 inserted by _Sean A. Irvine_, Apr 15 2026