 | ECS #38: Labelled 3-constrained functional graphs |
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1. Description
Labelled 3-constrained functional graphs
2. Specification
This labelled structure is specified as S in
\displaystyle
\left\{ S={\rm Set} \left( {\rm Cycle} \left( {\rm Prod} \left( Z,{
\rm Set} \left( g,{\rm card}=2 \right) \right) \right) \right) ,g={
\rm Union} \left( Z,{\rm Prod} \left( Z,{\rm Set} \left( g,{\rm card}=3
\right) \right) \right) \right\}
other formats
3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ \left( -117\,{n}^{2}-108\,n-54\,{n}^{3}-36-9\,{n}^{4}
\right) f \left( n \right) + \left( 8\,n+12 \right) f \left( n+3
\right) =0,f \left( 0 \right) =1,f \left( 1 \right) =0,f \left( 2
\right) =0 \right\}
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3.3. Closed form
\displaystyle
\cases{1/4\,{\frac {{3}^{1+5/3\,n} \left( \Gamma \left( 1/3\,n+2/3 \right) \right) ^{2} \left( \Gamma \left( 1/3\,n+1/3 \right) \right) ^{2}{2}^{-n}}{\Gamma \left( 1/3\,n+1/2 \right) {\pi }^{3/2}}}&${\rm irem} \left( n,3 \right) =0$\cr 0&${\rm irem} \left( n-1,3 \right) =0$\cr 0&${\rm irem} \left( n-2,3 \right) =0$\cr}
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3.4. Asymptotics
4. Exponential generating function
\displaystyle
-2\, \left( -2+x \left( {\rm RootOf} \left( -6\,{\rm \_Z}+6\,x+x{{\rm
\_Z}}^{3} \right) \right) ^{2} \right) ^{-1}
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It satisfies the following differential equation of order 2:
\displaystyle
\left\{ 18\,{x}^{2}y \left( x \right) + \left( 36\,{x}^{3}+4 \right) {
\frac {d}{dx}}y \left( x \right) + \left( 9\,{x}^{4}-8\,x \right) {
\frac {d^{2}}{d{x}^{2}}}y \left( x \right) =0,y \left( 0 \right) =1
\right\}
other formats
5. References
6. Random structure
Search a combinatorial structure by: (firstTerms should be a sequence of integers, separated by commas).