 | ECS #568: A simple regular expression in a labelled universe |
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1. Description
No description available
2. Specification
This labelled structure is specified as S in
\displaystyle
\left\{ S={\rm Sequence} \left( {\rm Prod} \left( {\rm Union} \left( Z
,Z \right) ,{\rm Sequence} \left( {\rm Prod} \left( Z,Z \right)
\right) \right) \right) \right\}
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3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ \left( -{n}^{2}-3\,n-2 \right) f \left( n \right) + \left( -2
\,n-4 \right) f \left( n+1 \right) +f \left( n+2 \right) =0,f \left( 0
\right) =1,f \left( 1 \right) =2,f \left( 2 \right) =8 \right\}
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3.3. Closed form
\displaystyle
\left( \sum _{\alpha={\rm RootOf} \left( -1+{{\rm \_Z}}^{2}+2\,{\rm
\_Z} \right) }-1/2\, \left( -1+\alpha \right) {\alpha}^{-1-n}+
\cases{1&$n=0$\cr 0&otherwise\cr} \right) n!
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3.4. Asymptotics
4. Exponential generating function
\displaystyle
{\frac {-1+{x}^{2}}{-1+{x}^{2}+2\,x}}
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5. References
6. Random structure
Search a combinatorial structure by: (firstTerms should be a sequence of integers, separated by commas).