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A258399
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Number of 4n-length strings of balanced parentheses of exactly n different types that are introduced in ascending order.
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4
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1, 2, 98, 11880, 2432430, 714249900, 275335499824, 131928199603200, 75727786603836510, 50713478000403718500, 38843740303576863755100, 33508462196084294380001040, 32157574295254903735909896240, 33990046387543889224733323929120
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OFFSET
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0,2
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..250
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FORMULA
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a(n) = A253180(2n,n).
a(n) ~ c * d^n * n! / n^(5/2), where d = A256254 = 98.8248737517356857317..., c = 0.04120447463568595292374592925415728563265717672... . - Vaclav Kotesovec, Jun 01 2015
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EXAMPLE
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a(0) = 1: the empty string.
a(1) = 2: ()(), (()).
a(2) = A000108(4) * (2^3-1) = 14*7 = 98.
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MAPLE
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ctln:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
A:= proc(n, k) option remember; k^n*ctln(n) end:
a:= n-> add(A(2*n, n-i)*(-1)^i/((n-i)!*i!), i=0..n):
seq(a(n), n=0..15);
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CROSSREFS
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Cf. A000108, A253180, A256254, A258426.
Sequence in context: A317729 A223038 A324266 * A212838 A024241 A278684
Adjacent sequences: A258396 A258397 A258398 * A258400 A258401 A258402
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KEYWORD
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nonn
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AUTHOR
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Alois P. Heinz, May 28 2015
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STATUS
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approved
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