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Factorial coefficient simplification

  1. Mar 28, 2012 #1

    I've been trying to find a way to simplify these two series coefficients,

    & {{a}_{4n}}={{a}_{o}}\frac{2!6!....(n-6)!(n-2)}{4!8!....(n-4)!n!} \\
    & {{a}_{4n+1}}={{a}_{1}}\frac{3!7!....(n-6)!(n-2)}{5!9!....(n-4)!n!} \\
    & E.g. \\
    & {{a}_{12}}={{a}_{o}}\frac{2!6!10!}{4!8!12!}\,= \frac{{{a}_{o}}}{3\cdot 4\cdot 7\cdot 8\cdot 11\cdot 12}\,\,\,\,\,\,and \\
    & {{a}_{13}}={{a}_{1}}\frac{3!7!11!}{5!9!13!}\,=\,\,\frac{{{a}_{1}}}{4\cdot 5\cdot 8\cdot 9\cdot 12\cdot 13} \\

    This is the best way I could find but I was wondering if there were any ways to avoid the use of ....'s

    Is there any way to simplify these coefficients?

    Thanks in advance
  2. jcsd
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