Av(13452, 13524, 13542, 14352, 14523, 14532, 15234, 15243, 15324, 15342, 15423, 15432, 51234, 51243, 51324, 51342, 51423, 51432)
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Generating Function
\(\displaystyle \frac{\left(x -1\right) \left(3 x^{6}-6 x^{5}+4 x^{4}-12 x^{3}+15 x^{2}-7 x +1\right)}{9 x^{7}-12 x^{6}+14 x^{5}-32 x^{4}+44 x^{3}-29 x^{2}+9 x -1}\)
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Counting Sequence
1, 1, 2, 6, 24, 102, 433, 1825, 7657, 32058, 134107, 560843, 2345284, 9807156, 41010189, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(9 x^{7}-12 x^{6}+14 x^{5}-32 x^{4}+44 x^{3}-29 x^{2}+9 x -1\right) F \! \left(x \right)-\left(x -1\right) \left(3 x^{6}-6 x^{5}+4 x^{4}-12 x^{3}+15 x^{2}-7 x +1\right) = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 102\)
\(\displaystyle a(6) = 433\)
\(\displaystyle a(7) = 1825\)
\(\displaystyle a{\left(n + 7 \right)} = 9 a{\left(n \right)} - 12 a{\left(n + 1 \right)} + 14 a{\left(n + 2 \right)} - 32 a{\left(n + 3 \right)} + 44 a{\left(n + 4 \right)} - 29 a{\left(n + 5 \right)} + 9 a{\left(n + 6 \right)}, \quad n \geq 8\)
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Explicit Closed Form
\(\displaystyle -\frac{1336476348 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +5}}{1242744671}-\frac{1336476348 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +5}}{1242744671}-\frac{1336476348 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +5}}{1242744671}-\frac{1336476348 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +5}}{1242744671}-\frac{1336476348 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +5}}{1242744671}-\frac{1336476348 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +5}}{1242744671}-\frac{1336476348 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +5}}{1242744671}+\frac{511619211 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +4}}{1242744671}+\frac{511619211 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +4}}{1242744671}+\frac{511619211 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +4}}{1242744671}+\frac{511619211 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +4}}{1242744671}+\frac{511619211 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +4}}{1242744671}+\frac{511619211 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +4}}{1242744671}+\frac{511619211 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +4}}{1242744671}-\frac{1394874942 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +3}}{1242744671}-\frac{1394874942 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +3}}{1242744671}-\frac{1394874942 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +3}}{1242744671}-\frac{1394874942 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +3}}{1242744671}-\frac{1394874942 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +3}}{1242744671}-\frac{1394874942 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +3}}{1242744671}-\frac{1394874942 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +3}}{1242744671}+\frac{3496220935 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +2}}{1242744671}+\frac{3496220935 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +2}}{1242744671}+\frac{3496220935 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +2}}{1242744671}+\frac{3496220935 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +2}}{1242744671}+\frac{3496220935 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +2}}{1242744671}+\frac{3496220935 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +2}}{1242744671}+\frac{3496220935 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +2}}{1242744671}-\frac{2996478561 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +1}}{1242744671}-\frac{2996478561 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +1}}{1242744671}-\frac{2996478561 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +1}}{1242744671}-\frac{2996478561 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +1}}{1242744671}-\frac{2996478561 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +1}}{1242744671}-\frac{2996478561 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +1}}{1242744671}-\frac{2996478561 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +1}}{1242744671}-\frac{108845404 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n -1}}{1242744671}-\frac{108845404 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n -1}}{1242744671}-\frac{108845404 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n -1}}{1242744671}-\frac{108845404 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n -1}}{1242744671}-\frac{108845404 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n -1}}{1242744671}-\frac{108845404 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n -1}}{1242744671}-\frac{108845404 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n -1}}{1242744671}+\frac{1092174886 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n}}{1242744671}+\frac{1092174886 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n}}{1242744671}+\frac{1092174886 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n}}{1242744671}+\frac{1092174886 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n}}{1242744671}+\frac{1092174886 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n}}{1242744671}+\frac{1092174886 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n}}{1242744671}+\frac{1092174886 \mathit{RootOf} \left(9 Z^{7}-12 Z^{6}+14 Z^{5}-32 Z^{4}+44 Z^{3}-29 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n}}{1242744671}+\left(\left\{\begin{array}{cc}\frac{1}{3} & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
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This specification was found using the strategy pack "Regular Insertion Encoding Left" and has 82 rules.

Finding the specification took 107 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)+F_{73}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{7}\! \left(x \right) &= 0\\ F_{8}\! \left(x \right) &= F_{16}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{17}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{16}\! \left(x \right) &= x\\ F_{17}\! \left(x \right) &= F_{16}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{16}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{29}\! \left(x \right)+F_{30}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{16}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{29}\! \left(x \right) &= 0\\ F_{30}\! \left(x \right) &= F_{16}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{16}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{16}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{16}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{65}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{16}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{56}\! \left(x \right)+F_{57}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{16}\! \left(x \right) F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{56}\! \left(x \right) &= 0\\ F_{57}\! \left(x \right) &= F_{16}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{16}\! \left(x \right) F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{16}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{16}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{16}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{7}\! \left(x \right)+F_{78}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{16}\! \left(x \right) F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{81}\! \left(x \right) &= 0\\ \end{align*}\)