PermPal Database

Generating Function

\frac{- 5 \sqrt{- 4 x + 1} x - 2 x^{2} + \sqrt{- 4 x + 1} + 13 x - 3}{2 x^{2} + 8 x - 2}

Copy to clipboard: Search on PermPAL

Counting Sequence

1, 1, 2, 6, 21, 79, 309, 1237, 5026, 20626, 85242, 354080, 1476368, 6173634, 25873744, ...

Search on OEIS Search on PermPAL

Implicit Equation for the Generating Function

(x^{2} + 4 x - 1) F {(x)}^{2} + (2 x^{2} - 13 x + 3) F (x) + x^{2} + 8 x - 2 = 0

Copy to clipboard: Search on PermPAL

Recurrence

a (0) = 1

a (1) = 1

a (2) = 2

a (3) = 6

a (4) = 21

a (n + 4) = \frac{10 (1 + 2 n) a (n)}{n + 4} + \frac{(20 + 71 n) a (n + 1)}{n + 4} - \frac{(86 + 55 n) a (n + 2)}{n + 4} + \frac{(36 + 13 n) a (n + 3)}{n + 4}, n \geq 5

Copy to clipboard:

Found on July 23, 2021.

Finding the specification took 11 seconds.

Copy to clipboard:

+ Expand

Copy 23 equations to clipboard:

\begin{aligned} F_{0} (x) & = F_{1} (x) + F_{2} (x) \\ F_{1} (x) & = 1 \\ F_{2} (x) & = F_{3} (x) F_{9} (x) \\ F_{3} (x) & = F_{1} (x) + F_{10} (x) + F_{4} (x) \\ F_{4} (x) & = F_{5} (x) \\ F_{5} (x) & = F_{3} (x) F_{6} (x) F_{9} (x) \\ F_{6} (x) & = F_{1} (x) + F_{7} (x) \\ F_{7} (x) & = F_{8} (x) \\ F_{8} (x) & = F_{6} {(x)}^{2} F_{9} (x) \\ F_{9} (x) & = x \\ F_{10} (x) & = F_{11} (x) F_{9} (x) \\ F_{11} (x) & = F_{12} (x, 1) \\ F_{12} (x, y) & = F_{1} (x) + F_{13} (x, y) + F_{15} (x, y) + F_{21} (x, y) \\ F_{13} (x, y) & = F_{12} (x, y) F_{14} (x, y) \\ F_{14} (x, y) & = y x \\ F_{15} (x, y) & = F_{16} (x, y) \\ F_{16} (x, y) & = F_{17} (x, y) F_{3} (x) F_{9} (x) \\ F_{17} (x, y) & = F_{1} (x) + F_{18} (x, y) + F_{19} (x, y) \\ F_{18} (x, y) & = F_{14} (x, y) F_{17} (x, y) \\ F_{19} (x, y) & = F_{20} (x, y) \\ F_{20} (x, y) & = F_{17} (x, y) F_{6} (x) F_{9} (x) \\ F_{21} (x, y) & = F_{22} (x, y) F_{9} (x) \\ F_{22} (x, y) & = \frac{y F_{12} (x, y) - F_{12} (x, 1)}{- 1 + y} \end{aligned}

Proof Tree