Av(12453, 12543, 21453, 21543, 24153, 24513, 25143, 25413)
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Counting Sequence
1, 1, 2, 6, 24, 112, 568, 3024, 16612, 93276, 532320, 3076440, 17960372, 105730364, 626798512, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{4}-4 x^{3}+5 x^{2}-4 x +2\right) F \left(x \right)^{4}+\left(7 x^{3}-17 x^{2}+18 x -10\right) F \left(x \right)^{3}+\left(12 x^{2}-24 x +18\right) F \left(x \right)^{2}+\left(10 x -14\right) F \! \left(x \right)+4 = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 112\)
\(\displaystyle a(6) = 568\)
\(\displaystyle a(7) = 3024\)
\(\displaystyle a(8) = 16612\)
\(\displaystyle a(9) = 93276\)
\(\displaystyle a(10) = 532320\)
\(\displaystyle a(11) = 3076440\)
\(\displaystyle a(12) = 17960372\)
\(\displaystyle a(13) = 105730364\)
\(\displaystyle a(14) = 626798512\)
\(\displaystyle a(15) = 3738195608\)
\(\displaystyle a(16) = 22410901332\)
\(\displaystyle a(17) = 134972816236\)
\(\displaystyle a{\left(n + 18 \right)} = \frac{2457 \left(n + 1\right) \left(n + 2\right) \left(n + 3\right) a{\left(n \right)}}{8 \left(n + 16\right) \left(n + 17\right) \left(n + 18\right)} - \frac{27 \left(n + 2\right) \left(n + 3\right) \left(1335 n + 4699\right) a{\left(n + 1 \right)}}{8 \left(n + 16\right) \left(n + 17\right) \left(n + 18\right)} + \frac{9 \left(n + 3\right) \left(19981 n^{2} + 161975 n + 331664\right) a{\left(n + 2 \right)}}{8 \left(n + 16\right) \left(n + 17\right) \left(n + 18\right)} + \frac{\left(29 n + 447\right) a{\left(n + 17 \right)}}{n + 18} - \frac{\left(712 n^{2} + 21191 n + 157722\right) a{\left(n + 16 \right)}}{2 \left(n + 17\right) \left(n + 18\right)} + \frac{\left(9623 n^{3} + 414993 n^{2} + 5964820 n + 28576542\right) a{\left(n + 15 \right)}}{4 \left(n + 16\right) \left(n + 17\right) \left(n + 18\right)} - \frac{\left(39043 n^{3} + 1585665 n^{2} + 21472097 n + 96955884\right) a{\left(n + 14 \right)}}{4 \left(n + 16\right) \left(n + 17\right) \left(n + 18\right)} + \frac{\left(96032 n^{3} - 379494 n^{2} - 35788415 n - 223758570\right) a{\left(n + 10 \right)}}{16 \left(n + 16\right) \left(n + 17\right) \left(n + 18\right)} - \frac{3 \left(102243 n^{3} + 3771475 n^{2} + 46165095 n + 187727336\right) a{\left(n + 12 \right)}}{8 \left(n + 16\right) \left(n + 17\right) \left(n + 18\right)} + \frac{\left(196559 n^{3} + 7544622 n^{2} + 96527989 n + 411747864\right) a{\left(n + 13 \right)}}{8 \left(n + 16\right) \left(n + 17\right) \left(n + 18\right)} - \frac{\left(353299 n^{3} + 4893588 n^{2} + 22930553 n + 35994540\right) a{\left(n + 3 \right)}}{8 \left(n + 16\right) \left(n + 17\right) \left(n + 18\right)} + \frac{\left(467305 n^{3} + 18371355 n^{2} + 230492282 n + 936832956\right) a{\left(n + 11 \right)}}{16 \left(n + 16\right) \left(n + 17\right) \left(n + 18\right)} + \frac{\left(478769 n^{3} + 8273982 n^{2} + 50887063 n + 105084210\right) a{\left(n + 4 \right)}}{32 \left(n + 16\right) \left(n + 17\right) \left(n + 18\right)} + \frac{3 \left(630305 n^{3} + 9754982 n^{2} + 37491381 n - 1217762\right) a{\left(n + 9 \right)}}{32 \left(n + 16\right) \left(n + 17\right) \left(n + 18\right)} - \frac{\left(4180985 n^{3} + 74144124 n^{2} + 435083365 n + 841327344\right) a{\left(n + 6 \right)}}{16 \left(n + 16\right) \left(n + 17\right) \left(n + 18\right)} + \frac{\left(4269907 n^{3} + 77123406 n^{2} + 471299549 n + 979111590\right) a{\left(n + 5 \right)}}{32 \left(n + 16\right) \left(n + 17\right) \left(n + 18\right)} + \frac{\left(4575514 n^{3} + 79382985 n^{2} + 438654482 n + 744991554\right) a{\left(n + 7 \right)}}{16 \left(n + 16\right) \left(n + 17\right) \left(n + 18\right)} - \frac{\left(5971595 n^{3} + 98990334 n^{2} + 477395227 n + 525021234\right) a{\left(n + 8 \right)}}{32 \left(n + 16\right) \left(n + 17\right) \left(n + 18\right)}, \quad n \geq 18\)
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This specification was found using the strategy pack "Point Placements Req Corrob" and has 46 rules.

Finding the specification took 568 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_{26}\! \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_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{26}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{13}\! \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_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{11}\! \left(x \right) &= \frac{F_{12}\! \left(x \right)}{F_{26}\! \left(x \right)}\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{19}\! \left(x \right) &= \frac{F_{20}\! \left(x \right)}{F_{26}\! \left(x \right)}\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= -F_{13}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= \frac{F_{23}\! \left(x \right)}{F_{26}\! \left(x \right)}\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{26}\! \left(x \right) &= x\\ F_{27}\! \left(x \right) &= F_{0}\! \left(x \right) F_{28}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{29}\! \left(x \right) x +F_{29} \left(x \right)^{2}+x\\ F_{30}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= -F_{34}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= \frac{F_{33}\! \left(x \right)}{F_{26}\! \left(x \right)}\\ F_{33}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{34}\! \left(x \right) x +F_{34} \left(x \right)^{2}-2 F_{34}\! \left(x \right)+2\\ F_{35}\! \left(x \right) &= \frac{F_{36}\! \left(x \right)}{F_{26}\! \left(x \right) F_{34}\! \left(x \right)}\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= -F_{34}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= \frac{F_{39}\! \left(x \right)}{F_{0}\! \left(x \right)}\\ F_{39}\! \left(x \right) &= -F_{42}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= \frac{F_{41}\! \left(x \right)}{F_{26}\! \left(x \right)}\\ F_{41}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{42}\! \left(x \right) &= -F_{45}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= \frac{F_{44}\! \left(x \right)}{F_{26}\! \left(x \right)}\\ F_{44}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{2}\! \left(x \right) F_{38}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Point Placements Req Corrob" and has 44 rules.

Finding the specification took 1053 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_{25}\! \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_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{25}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{13}\! \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_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{11}\! \left(x \right) &= \frac{F_{12}\! \left(x \right)}{F_{25}\! \left(x \right)}\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{19}\! \left(x \right) &= \frac{F_{20}\! \left(x \right)}{F_{25}\! \left(x \right)}\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= -F_{13}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= \frac{F_{23}\! \left(x \right)}{F_{25}\! \left(x \right)}\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{25}\! \left(x \right) &= x\\ F_{26}\! \left(x \right) &= F_{0}\! \left(x \right) F_{27}\! \left(x \right) F_{30}\! \left(x \right)\\ F_{27}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= \frac{F_{29}\! \left(x \right)}{F_{25}\! \left(x \right)}\\ F_{29}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{30}\! \left(x \right) &= \frac{F_{31}\! \left(x \right)}{F_{25}\! \left(x \right) F_{43}\! \left(x \right)}\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= -F_{43}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= \frac{F_{34}\! \left(x \right)}{F_{0}\! \left(x \right)}\\ F_{34}\! \left(x \right) &= -F_{37}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= \frac{F_{36}\! \left(x \right)}{F_{25}\! \left(x \right)}\\ F_{36}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{37}\! \left(x \right) &= -F_{42}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= \frac{F_{39}\! \left(x \right)}{F_{25}\! \left(x \right)}\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= -F_{41}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{41}\! \left(x \right) x +F_{41} \left(x \right)^{2}+x\\ F_{42}\! \left(x \right) &= F_{2}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{43}\! \left(x \right) x +F_{43} \left(x \right)^{2}-2 F_{43}\! \left(x \right)+2\\ \end{align*}\)