Gourevitch's series identity

References:

• About a New Kind of Ramanujan-Type Series by Jesús Guillera
• [G2003] Guillera, Jesús. "About a new kind of Ramanujan-type series." Experimental Mathematics 12.4 (2003): 507-510.
• [A2025] Au, Kam Cheong. "Wilf-Zeilberger seeds and non-trivial hypergeometric identities." Journal of Symbolic Computation 130 (2025): 102421. arXiv:2312.14051

theorem Gourevitch.gourevitch_series_identity : ∑' (n : ℕ), (1 + 14 * ↑n + 76 * ↑n ^ 2 + 168 * ↑n ^ 3) / 2 ^ (20 * n) * ↑n.centralBinom ^ 7 = 32 / Real.pi ^ 3

The Gourevitch series identity: The following idenitity holds: $\sum_{n=0}^{\infty} \frac{1 + 14 n + 76 n^2 + 168 n^3}{2^{20 n}} \binom{2n}{n}^7 = \frac{32}{\pi^3}.$ This was originally conjectured in [G2003] by Guillera and proven in [A2025] by Au.