PermPal Database

Av(1324, 1342, 2143)

Generating Function

\frac{- 2 \sqrt{- 4 x + 1} x + \sqrt{- 4 x + 1} - 4 x + 1}{8 x^{2} - 10 x + 2}

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Counting Sequence

1, 1, 2, 6, 21, 77, 287, 1079, 4082, 15522, 59280, 227240, 873886, 3370030, 13027730, ...

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Implicit Equation for the Generating Function

{(x - 1)}^{2} (4 x - 1) F {(x)}^{2} + (x - 1) (4 x - 1) F (x) + x^{2} = 0

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Recurrence

a (0) = 1

a (1) = 1

a (2) = 2

a (n + 3) = \frac{4 (1 + 2 n) a (n)}{n + 3} - \frac{2 (8 + 7 n) a (n + 1)}{n + 3} + \frac{(15 + 7 n) a (n + 2)}{n + 3}, n \geq 3

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This specification was found using the strategy pack "Point Placements" and has 20 rules.

Found on July 23, 2021.

Finding the specification took 3 seconds.

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\begin{aligned} F_{0} (x) & = F_{1} (x) + F_{2} (x) \\ F_{1} (x) & = 1 \\ F_{2} (x) & = F_{3} (x) \\ F_{3} (x) & = F_{10} (x) F_{4} (x) \\ F_{4} (x) & = F_{0} (x) + F_{5} (x) \\ F_{5} (x) & = F_{6} (x) \\ F_{6} (x) & = F_{10} (x) F_{11} (x) F_{7} (x) \\ F_{7} (x) & = F_{1} (x) + F_{8} (x) \\ F_{8} (x) & = F_{9} (x) \\ F_{9} (x) & = F_{7} {(x)}^{2} F_{10} (x) \\ F_{10} (x) & = x \\ F_{11} (x) & = F_{12} (x) + F_{13} (x) \\ F_{12} (x) & = F_{0} (x) F_{7} (x) \\ F_{13} (x) & = F_{14} (x) F_{17} (x) \\ F_{14} (x) & = F_{1} (x) + F_{15} (x) \\ F_{15} (x) & = F_{16} (x) \\ F_{16} (x) & = F_{10} (x) F_{14} (x) \\ F_{17} (x) & = - F_{0} (x) + F_{18} (x) \\ F_{18} (x) & = \frac{F_{19} (x)}{F_{10} (x)} \\ F_{19} (x) & = F_{2} (x) \end{aligned}

Av(1324, 1342, 2143)

This specification was found using the strategy pack "Point Placements" and has 20 rules.

Proof Tree

System of Equations