0,2

In nondeterministic walks (N-walks) the steps are sets and called N-steps. N-walks start at 0 and are concatenations of such N-steps such that all possible extensions are explored in parallel. The nondeterministic Dyck step set is { {-1}, {1}, {-1,1} }. Such an N-walk is called an N-bridge if it contains at least one trajectory that is a classical bridge, i.e., starts and ends at 0 (for more details see the de Panafieu-Wallner article).

Élie de Panafieu and Michael Wallner, Combinatorics of nondeterministic walks, arXiv:2311.03234 [math.CO], 2023.

G.f.: (1-6*t)/(sqrt(1-8*t)*(1-9*t)).

From Joseph M. Shunia, May 09 2024: (Start)

a(n) = A089022(n) + Sum_{k=0..n-1} binomial(2*n, k)*2^(2*n-k).

a(n) = A000244(2*n) - Sum_{k=n+1..2*n} binomial(2*n, k)*2^(2*n-k+1). (End)

The a(1)=7 N-bridges of length 2 are

/ / /

/\, , /\, , /\, / , /\

\/ \/ \ \/ \/

\ \ \

Cf. A151281 (nondeterministic Dyck meanders), A368234 (nondeterministic Dyck excursions), A000244 (nondeterministic Dyck walks).

Sequence in context: A155132 A270472 A266426 * A345078 A380686 A233743

Adjacent sequences: A368161 A368162 A368163 * A368165 A368166 A368167

Michael Wallner, Dec 14 2023

approved