K (ngn/k), 36 35 bytes

{"()"(&/-1<+\1-2*)#(x=+/)#+!2|&2*x}

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+!2|&2*x all binary vectors of length 2*n

(x=+/)# only those that sum to n

(&/-1<+\1-2*)# only those whose partial sums, treating 0/1 as 1/-1, are nowhere negative

"()" use 0/1 as indices in this string

K (ngn/k), 36 35 37 bytes

{"()"(|/'0>+\'1-2*)_(x=+/')#+!2|&2*x}

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+!2|&2*x all binary vectors of length 2*n

(x=+/')# only those that sum to n

(|/0>+\'1-2*)_ only those whose partial sums, treating 0/1 as 1/-1, are nowhere negative

"()" use 0/1 as indices in this string

K (ngn/k), 36 bytes

{"()"(&/-1<+\1-2*)#(x=+/)#+!(2*x)#2}

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+!(2*x)#2 all binary vectors of length 2*n

(x=+/)# only those that sum to n

(&/-1<+\1-2*)# only those whose partial sums, treating 0/1 as 1/-1, are nowhere negative

"()" use 0/1 as indices in this string

K (ngn/k), 36 35 bytes

{"()"(&/-1<+\1-2*)#(x=+/)#+!2|&2*x}

Try it online!

+!2|&2*x all binary vectors of length 2*n

(x=+/)# only those that sum to n

(&/-1<+\1-2*)# only those whose partial sums, treating 0/1 as 1/-1, are nowhere negative

"()" use 0/1 as indices in this string

K (ngn/k), 36 bytes

{"()"(&/-1<+\1-2*)#(x=+/)#+!(2*x)#2}

Try it online!

K (ngn/k), 36 bytes

{"()"(&/-1<+\1-2*)#(x=+/)#+!(2*x)#2}

Try it online!

+!(2*x)#2 all binary vectors of length 2*n

(x=+/)# only those that sum to n

(&/-1<+\1-2*)# only those whose partial sums, treating 0/1 as 1/-1, are nowhere negative

"()" use 0/1 as indices in this string