 | ECS #459: A simple regular expression |
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1. Description
No description available
2. Specification
This unlabelled structure is specified as S in
\displaystyle
\left\{ S={\rm Sequence} \left( {\rm Prod} \left( Z,{\rm Sequence}
\left( Z \right) ,{\rm Sequence} \left( Z \right) ,{\rm Sequence}
\left( Z \right) \right) \right) \right\}
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3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ f \left( n \right) -3\,f \left( n+1 \right) +4\,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) =4,f \left( 3 \right) =13 \right\}
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3.3. Closed form
\displaystyle
\sum _{\alpha={\rm RootOf} \left( -1+4\,{\rm \_Z}-3\,{{\rm \_Z}}^{2}+{{
\rm \_Z}}^{3} \right) }-1/31\, \left( 5\,\alpha+3\,{\alpha}^{2}-6
\right) {\alpha}^{-1-n}+\cases{1&$n=0$\cr 0&otherwise\cr}
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3.4. Asymptotics
4. Ordinary generating function
\displaystyle
{\frac { \left( -1+x \right) ^{3}}{-1+4\,x-3\,{x}^{2}+{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).