 | ECS #598: 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( Z,Z,Z,{\rm Union}
\left( Z,{\rm Sequence} \left( Z \right) \right) \right) \right)
\right\}
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3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ \left( {n}^{5}+15\,{n}^{4}+85\,{n}^{3}+225\,{n}^{2}+274\,n+
120 \right) f \left( n \right) + \left( -{n}^{4}-14\,{n}^{3}-71\,{n}^{2
}-154\,n-120 \right) f \left( n+1 \right) + \left( -{n}^{3}-12\,{n}^{2}
-47\,n-60 \right) f \left( n+2 \right) + \left( -n-5 \right) f \left( n
+4 \right) +f \left( n+5 \right) =0,f \left( 0 \right) =1,f \left( 1
\right) =0,f \left( 2 \right) =0,f \left( 3 \right) =6,f \left( 4
\right) =48 \right\}
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3.3. Closed form
\displaystyle
\sum _{\alpha={\rm RootOf} \left( 1-{\rm \_Z}-{{\rm \_Z}}^{4}+{{\rm \_Z
}}^{5}-{{\rm \_Z}}^{3} \right) }{\frac {1}{8519}}\, \left( 138+2003\,
\alpha-346\,{\alpha}^{2}-444\,{\alpha}^{3}+11\,{\alpha}^{4} \right) {
\alpha}^{-1-n}n!
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3.4. Asymptotics
4. Exponential generating function
\displaystyle
-{\frac {-1+x}{1-x-{x}^{4}+{x}^{5}-{x}^{3}}}
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5. References
6. Random structure
Search a combinatorial structure by: (firstTerms should be a sequence of integers, separated by commas).