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A052734
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a(n) = 4^(n-1) * n! * Catalan(n-1) for n > 0, with a(0) = 0.
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8
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0, 1, 8, 192, 7680, 430080, 30965760, 2724986880, 283398635520, 34007836262400, 4625065731686400, 703009991216332800, 118105678524343910400, 21731444848479279513600, 4346288969695855902720000, 938798417454304874987520000, 217801232849398730997104640000, 54014705746650885287281950720000, 14259882317115833715842434990080000
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OFFSET
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0,3
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COMMENTS
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For n>0, the number of fully-parenthesized expressions that you can form with n operands and 4 types of binary operators. - Yogy Namara (yogy.namara(AT)gmail.com), Mar 07 2010
a(n+1) is the number of square roots of any permutation in S_{16*n} whose disjoint cycle decomposition consists of 2*n cycles of length 8. - Luis Manuel Rivera Martínez, Feb 26 2015
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LINKS
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FORMULA
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E.g.f.: (1 - sqrt(1-16*x))/8.
Recurrence: a(1)=1, 8*(1 - 2*n)*a(n) + a(n+1) = 0.
a(n) = 16^n*Gamma(n+1/2)/sqrt(Pi).
a(0) = 0, a(1) = 1; a(n) = 4 * Sum_{k=1..n-1} binomial(n,k) * a(k) * a(n-k). - Ilya Gutkovskiy, Jul 09 2020
Sum_{n>=1} 1/a(n) = 1 + e^(1/16)*sqrt(Pi)*erf(1/4)/4, where erf is the error function.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - e^(-1/16)*sqrt(Pi)*erfi(1/4)/4, where erfi is the imaginary error function. (End)
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EXAMPLE
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Let's say the 4 types of binary operators are +, -, *, and /. Then, with 3 operands {a, b, c}, we can form expressions such as ((b+a)/c), (a-(c-b)), (c*(b+a)), etc. There are a(3)=192 such expressions. - Yogy Namara (yogy.namara(AT)gmail.com), Mar 07 2010
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MAPLE
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spec := [S, {B=Prod(C, C), S=Union(B, Z), C=Union(B, S, Z)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
seq((2*n)!/n! * 4^n, n = 0..10);
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MATHEMATICA
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Join[{0}, Table[CatalanNumber[n-1] 4^(n-1) n!, {n, 1, 20}]] (* Vincenzo Librandi, Mar 11 2013 *)
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PROG
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(Magma) [0] cat [Catalan(n-1)*4^(n-1)*Factorial(n): n in [1..20]]; // Vincenzo Librandi, Mar 11 2013
(Sage) [0]+[4^(n-1)*factorial(n)*catalan_number(n-1) for n in (1..30)] # G. C. Greubel, Apr 02 2021
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CROSSREFS
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Equal to A000108 if all operands and all operators are indistinguishable.
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KEYWORD
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easy,nonn
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AUTHOR
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encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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EXTENSIONS
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STATUS
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approved
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