 | ECS #611: 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,{\rm Union}
\left( Z,Z,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( 3\,{n}^{3}+18\,{n}^{2}+33\,n+18 \right) f \left( n
\right) + \left( -3\,{n}^{2}-15\,n-18 \right) f \left( n+1 \right) +
\left( -6-2\,n \right) f \left( n+2 \right) +f \left( n+3 \right) =0,f
\left( 0 \right) =1,f \left( 1 \right) =1,f \left( 2 \right) =10
\right\}
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3.3. Closed form
\displaystyle
\sum _{\alpha={\rm RootOf} \left( 1-2\,{\rm \_Z}-3\,{{\rm \_Z}}^{2}+3\,
{{\rm \_Z}}^{3} \right) }-{\frac {1}{107}}\, \left( -13-38\,\alpha+33\,
{\alpha}^{2} \right) {\alpha}^{-n-1}n!
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3.4. Asymptotics
4. Exponential generating function
\displaystyle
-{\frac {-1+x}{1-2\,x-3\,{x}^{2}+3\,{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).