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A221954
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a(n) = 3^(n-1) * n! * Catalan(n-1).
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9
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1, 6, 108, 3240, 136080, 7348320, 484989120, 37829151360, 3404623622400, 347271609484800, 39588963481267200, 4988209398639667200, 688372897012274073600, 103255934551841111040000, 16727461397398259988480000, 2910578283147297237995520000, 541367560665397286267166720000, 107190777011748662680899010560000, 22510063172467219162988792217600000
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OFFSET
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1,2
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COMMENTS
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a(n+1) is the number of square roots of any permutation in S_{12*n} whose disjoint cycle decomposition consists of 2*n cycles of length 6. - Luis Manuel Rivera Martínez, Feb 26 2015
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LINKS
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FORMULA
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a(n) = 12^(n-1) * Gamma(n - 1/2) / sqrt(Pi). - Daniel Suteu, Jan 06 2017
a(1) = 1; a(n) = 3 * 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^(1/12)*sqrt(Pi)*erf(1/(2*sqrt(3)))/(2*sqrt(3)), where erf is the error function.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - e^(-1/12)*sqrt(Pi)*erfi(1/(2*sqrt(3)))/(2*sqrt(3)), where erfi is the imaginary error function. (End)
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MAPLE
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MATHEMATICA
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PROG
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(Magma) [Catalan(n-1)*3^(n-1)*Factorial(n): n in [1..20]]; // Vincenzo Librandi, Mar 11 2013
(PARI) my(x='x+O('x^22)); Vec(serlaplace((1-sqrt(1-12*x))/6)) \\ Michel Marcus, Mar 04 2015
(Sage) [3^(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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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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