**I. Recurrences**
In a [previous post][1], it was mentioned how Almkvist-Zudilin did a computer search for solutions to the recurrence relation,
$$(n+1)^3s_{n+1}=(2n+1)(an^2+an+b)s_n+c\,n^3s_{n-1}$$
***within a bound*** and found 6 non-trivial solutions (essentially related to Zagier's 6 sporadic sequences). But if they used $|c|<256$, then they wouldn't find the four recurrences below where it goes as high as $c = 432^2 = 186624$,
\begin{align}
(n+1)^3\alpha_{n+1}&=(2n+1)(432n^2+432n+312)\alpha_n-432^2n^3\alpha_{n-1}\\[6pt]
(n+1)^3\beta_{n+1}&=(2n+1)(64n^2+64n+40)\beta_n-64^2n^3\beta_{n-1}\\[6pt]
(n+1)^3\gamma_{n+1}&=(2n+1)(27n^2+27n+15)\gamma_n-27^2n^3\gamma_{n-1}\\[6pt]
(n+1)^3\delta_{n+1}&=(2n+1)(16n^2+16n+8)\delta_n-16^2n^3\delta_{n-1}
\end{align}
While the recurrences found by Zagier, Cooper, et al had a modular interpretation for levels 5, 6, 7, etc, *these are for levels 1 to 4*. The associated sequences are below.
---
**II. Sequences**
Given the binomial $\binom{n}{m}$, then,
\begin{align}
\alpha(n) &= (-1)^n\sum_{j=0}^n(-432)^{n-j}\binom{n+j}{n-j}\binom{2j}j\binom{3j}j\binom{6j}{3j}\\
&= 1, 312, 114264, 44196288,\dots\\[6pt]
\beta(n) &= (-1)^n\sum_{j=0}^n(-64)^{n-j}\binom{n+j}{n-j}\binom{2j}{j}^2\binom{4j}{2j}\\
&=1, 40, 2008, 109120,\dots \\[6pt]
\gamma(n) &= (-1)^n\sum_{j=0}^n(-27)^{n-j}\binom{n+j}{n-j}\binom{2j}{j}^2\binom{3j}{j}\\
&=1, 15, 297, 6495,\dots\\[6pt]
\delta(n) &= (-1)^n\sum_{j=0}^n(-16)^{n-j}\binom{n+j}{n-j}\binom{2j}{j}^3\\
&=1, 8, 88, 1088,\dots\\
\end{align}
where all $s(0)=1.$ These can easily be derived (using [Method 1 in this post][2]) from Ramanujan's original four sequences for his 1/pi formulas so they are "Ramanujan-type". (*My thanks to Michael Somos for providing a Mathematica code to find recurrence relations.*)
Incidentally, the 2nd and 4th have "simpler" formulations. *Do the other two have as well?*
\begin{align}
\beta(n) &= \sum_{j=0}^n 16^{n-j}\binom{2j}{j}^3\binom{2n-2j}{n-j}\qquad\\
\delta(n) &= \sum_{j=0}^n\binom{2j}{j}^2\binom{2n-2j}{n-j}^2\qquad\\
\end{align}
---
**III. Modular context and pi formulas**
Each of these sequences are associated with a McKay-Thompson series of level 1,2,3,4. The first is connected to the $j$-function while the other three are for the eta quotients,
\begin{align}
j_{2B}(\tau) &= \left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{24}\\
j_{3B}(\tau) &= \left(\frac{\eta(\tau)}{\eta(3\tau)}\right)^{12}\\
j_{4C}(\tau) &= \left(\frac{\eta(\tau)}{\eta(4\tau)}\right)^{8}
\end{align}
For example, let $\tau=\frac12\sqrt{-58},\,$ so $\,j_{2B}(\tau)=64\left(\frac{5+\sqrt{29}}{2}\right)^{12}=D.$ Then,
$$\frac{1}{\pi} = 16\sqrt{2}\,\sum_{n=0}^\infty (-1)^n \,\beta(n)\,\frac{-24184+9801\sqrt{29}\,\left(n+\frac12\right)}{D^{n+\frac12}}$$
so the $\beta$ sequence can be used for a Ramanujan-type pi formula, just like Zagier's sporadics. For pi formulas using all four $\alpha, \beta, \gamma, \delta$, see [*Ramanujan-Sato series*][3].
---
**IV. Continued fractions**
The cfracs of the sporadics had closed-forms. Using the polynomials above (and up to $m = 12000$), Wolfram Alpha was accurate only to a few decimal places before timing out,
\begin{align}
F_1 &= \frac1{312 + \large{\underset{n=1}{\overset{m}{\mathrm K}} ~ \frac{-432^2 n^6}{(2n+1)(432n^2+432n+312)}}} = 0.0045793\dots\\
F_2 &= \frac1{40 + \large{\underset{n=1}{\overset{m}{\mathrm K}} ~ \frac{-64^2 n^6}{(2n+1)(64n^2+64n+40)}}} \;=\; 0.041425\dots\\
F_3 &= \frac1{15 + \large{\underset{n=1}{\overset{m}{\mathrm K}} ~ \frac{-27^2 n^6}{(2n+1)(27n^2+27n+15)}}} = 0.1366\dots\\
F_4 &= \frac1{8 + \large{\underset{n=1}{\overset{m}{\mathrm K}} ~ \frac{-16^2 n^6}{(2n+1)(16n^2+16n+8)}}} = 0.406\dots\\
\end{align}
with the last having the slowest "convergence". *What are these numbers?*
---
**V. Questions**
1. Starting with $s(-1) = 0, s(0)=1$, do the recurrences really yield integers for all $n$?
2. Are there simpler formulas for the other two sequences ($\alpha$ and $\gamma$)?
3. And are there closed-forms for the continued fractions $F_i$?
[1]: https://mathoverflow.net/q/447166/12905
[2]: https://mathoverflow.net/q/446778/12905
[3]: https://en.wikipedia.org/wiki/Ramanujan%E2%80%93Sato_series