Av(12345, 12354, 12435, 12453, 12534, 12543, 13425, 13524, 21345, 21354, 21435, 21453, 21534, 21543, 23415, 23514, 31245, 31254, 31425, 31452, 31524, 31542, 32415, 32514, 41235, 41253, 41325, 41352, 41523, 41532, 42315, 42513, 51234, 51243, 51324, 51342, 51423, 51432, 52314, 52413)
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Generating Function
\(\displaystyle \frac{4 x^{6}-10 x^{5}+14 x^{4}-9 x^{3}+9 x^{2}-5 x +1}{\left(2 x^{3}-2 x^{2}+3 x -1\right)^{2}}\)
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Counting Sequence
1, 1, 2, 6, 24, 80, 240, 700, 2016, 5724, 16040, 44484, 122352, 334204, 907480, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{3}-2 x^{2}+3 x -1\right)^{2} F \! \left(x \right)-4 x^{6}+10 x^{5}-14 x^{4}+9 x^{3}-9 x^{2}+5 x -1 = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 80\)
\(\displaystyle a(6) = 240\)
\(\displaystyle a{\left(n + 6 \right)} = - 4 a{\left(n \right)} + 8 a{\left(n + 1 \right)} - 16 a{\left(n + 2 \right)} + 16 a{\left(n + 3 \right)} - 13 a{\left(n + 4 \right)} + 6 a{\left(n + 5 \right)}, \quad n \geq 7\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ -\frac{11 \left(\left(\left(\left(\frac{182 i 2^{\frac{1}{3}}}{33}-\frac{7 \,2^{\frac{5}{6}} \sqrt{13}}{132}\right) \sqrt{3}-\frac{7 i 2^{\frac{5}{6}} \sqrt{13}}{44}+\frac{182 \,2^{\frac{1}{3}}}{33}\right) \left(4+3 \sqrt{13}\, \sqrt{2}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{1274}{33}+\left(\left(-\frac{13 i 2^{\frac{2}{3}}}{132}-\frac{\sqrt{13}\, 2^{\frac{1}{6}}}{3}\right) \sqrt{3}+i \sqrt{13}\, 2^{\frac{1}{6}}+\frac{13 \,2^{\frac{2}{3}}}{132}\right) \left(4+3 \sqrt{13}\, \sqrt{2}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(-49 i \sqrt{3}-49\right) \left(8+6 \sqrt{13}\, \sqrt{2}\, \sqrt{3}\right)^{\frac{1}{3}}}{588}+\frac{1}{3}+\frac{\left(-9 \,2^{\frac{1}{6}} \left(i-\frac{\sqrt{3}}{3}\right) \sqrt{13}+\left(2 i \sqrt{3}-2\right) 2^{\frac{2}{3}}\right) \left(4+3 \sqrt{13}\, \sqrt{2}\, \sqrt{3}\right)^{\frac{2}{3}}}{588}\right)^{-n}+\left(\left(\left(-\frac{182 i 2^{\frac{1}{3}}}{33}-\frac{7 \,2^{\frac{5}{6}} \sqrt{13}}{132}\right) \sqrt{3}+\frac{7 i 2^{\frac{5}{6}} \sqrt{13}}{44}+\frac{182 \,2^{\frac{1}{3}}}{33}\right) \left(4+3 \sqrt{13}\, \sqrt{2}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{1274}{33}+\left(\left(\frac{13 i 2^{\frac{2}{3}}}{132}-\frac{\sqrt{13}\, 2^{\frac{1}{6}}}{3}\right) \sqrt{3}-i \sqrt{13}\, 2^{\frac{1}{6}}+\frac{13 \,2^{\frac{2}{3}}}{132}\right) \left(4+3 \sqrt{13}\, \sqrt{2}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(49 i \sqrt{3}-49\right) \left(8+6 \sqrt{13}\, \sqrt{2}\, \sqrt{3}\right)^{\frac{1}{3}}}{588}+\frac{1}{3}+\frac{\left(9 \,2^{\frac{1}{6}} \left(i+\frac{\sqrt{3}}{3}\right) \sqrt{13}+\left(-2 i \sqrt{3}-2\right) 2^{\frac{2}{3}}\right) \left(4+3 \sqrt{13}\, \sqrt{2}\, \sqrt{3}\right)^{\frac{2}{3}}}{588}\right)^{-n}+\frac{2 \left(\left(\frac{7 \,2^{\frac{5}{6}} \sqrt{13}\, \sqrt{3}}{44}-\frac{182 \,2^{\frac{1}{3}}}{11}\right) \left(4+3 \sqrt{13}\, \sqrt{2}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{637}{11}+\left(2^{\frac{1}{6}} \sqrt{3}\, \sqrt{13}-\frac{13 \,2^{\frac{2}{3}}}{44}\right) \left(4+3 \sqrt{13}\, \sqrt{2}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(-3 \,2^{\frac{1}{6}} \sqrt{3}\, \sqrt{13}+2 \,2^{\frac{2}{3}}\right) \left(4+3 \sqrt{13}\, \sqrt{2}\, \sqrt{3}\right)^{\frac{2}{3}}}{294}+\frac{\left(8+6 \sqrt{13}\, \sqrt{2}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{1}{3}\right)^{-n}}{3}\right) n}{2548} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Point Placements" and has 67 rules.

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