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 A258399 Number of 4n-length strings of balanced parentheses of exactly n different types that are introduced in ascending order. 4
 1, 2, 98, 11880, 2432430, 714249900, 275335499824, 131928199603200, 75727786603836510, 50713478000403718500, 38843740303576863755100, 33508462196084294380001040, 32157574295254903735909896240, 33990046387543889224733323929120 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..250 FORMULA 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 EXAMPLE a(0) = 1: the empty string. a(1) = 2: ()(), (()). a(2) = A000108(4) * (2^3-1) = 14*7 = 98. MAPLE 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); MATHEMATICA A[n_, k_] := A[n, k] = k^n CatalanNumber[n]; a[n_] := If[n==0, 1, Sum[A[2n, n-i] (-1)^i/((n-i)! i!), {i, 0, n}]]; a /@ Range[0, 15] (* Jean-François Alcover, Jan 01 2021, after Alois P. Heinz *) CROSSREFS Cf. A000108, A253180, A256254, A258426. Sequence in context: A317729 A223038 A324266 * A212838 A024241 A278684 Adjacent sequences:  A258396 A258397 A258398 * A258400 A258401 A258402 KEYWORD nonn AUTHOR Alois P. Heinz, May 28 2015 STATUS approved

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Last modified May 14 22:40 EDT 2021. Contains 343909 sequences. (Running on oeis4.)