<s>[INST] Formulas for the polygonal or figurate numbers? Here is an interesting formula for the reciprocal of the heptagonal numbers. Are there any other analogous formulas for the polygonal or figurate numbers?
$$
\sum_{n=1}^\infty \frac{2}{n(5n-3)} =\\
\frac{1}{15}{\pi}{\sqrt{25-10\sqrt{5}}}+\frac{2}{3}\ln(5)+\frac{{1}+\sqrt{5}}{3}\ln\left(\frac{1}{2}\sqrt{10-2\sqrt{5}}\right)+\frac{{1}-\sqrt{5}}{3}\ln\left(\frac{1}{2}\sqrt{10+2\sqrt{5}}\right)
$$
Reference: "Beyond The Basel Problem" L. Downey ,B. Ong ,J. Sellers
 [/INST] figurate numbers 
$$ \sum_{n=1}^{\infty} \frac{1}{\dbinom{n+r-1}{r}} = \sum_{n=1}^{\infty} \frac{r!(n-1)!}{(n+r-1)!} $$ 
$$ = r\sum_{n=1}^{\infty} \frac{\Gamma(r) \Gamma(n)}{\Gamma(n +r )} = r\sum_{n=1}^{\infty} \beta(r,n) = r\sum_{n=1}^{\infty} \int_0^1 x^{n-1}(1 - x)^{r-1} \ dx = r\int_0^1 (1-x)^{r-1} \sum_{n=1}^{\infty} x^{n-1} \ dx $$
$$ = r\int_0^1 (1-x)^{r-2} \ dx = -r\left[ \frac{(1-x)^{r-1}}{r-1} \right]_0^1 = \frac{r}{r-1} $$
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