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# Central binomial sums research
Checked on 2026-11-11: CGLXZ arXiv:2210.14704v1, equations (1.1)-(1.2)
and Lemma 2.1 (2.1)-(2.3), supports the application arguments +/-1/16,
both signs, the initial term, and the order shift. Saved HTML and its hash
appear in evidence-sha256.json. DLMF 5.12.1, 5.5.5 and 5.7.3 were inspected
through the web reader; direct Python downloads returned HTTP 403. They
supply the beta integral, duplication and log-Gamma expansion used in the
positive-endpoint derivation in computation.md.
Checked catalogue: 711 records (including authenticated draft metadata).
already_here matched related families but no identical quantity; already_asked
failed through the anonymous search API, then authenticated gh search returned
only #318. api/lookup at pi/2, pi/3 and asinh(1) found other mathematical
contexts, not this family. Existing tags: number theory, special values,
special functions. binomial sums and polylogarithms are absent; retain these
as tag proposals, without modifying other tables.
Related: T698 is inverse-binomial; T697 holds log-sine moments; T177 is beta.
T713 holds log-sinh moments but is currently offered and not public. Its
planned relation is U_p(-1/16)=(-1)^p Lsh_p(2log(phi))/(p-1)!; do not add an
inaccessible public cross-link before it is published. The selected draft is
T715, Central_binomial_sums. The planned harmonic sibling has not been built
by this run. Preserve the n=0 term and the H_(2n+1) shift when it is built.
Both parameter selections are justified in argument-selection.md. No unresolved
source convention remains. The finite p<=9 cutoff is noncanonical and the
natural domain is larger. Numerical evidence is reproducible from generate.py,
checks.py and extra_checks.py. The latter compares companion conventions and
uses mpmath hypergeometric evaluation on all 18 interior cases; companions
are not independent numerical evidence. Both main algorithms use Arb.
Provenance checked: proposal #318 explicitly contributes to wanted #263;
the latter includes related hypergeometric/Feynman-integral contexts. This is
partial progress only. Both exact links were sent on initial draft creation
and retained on subsequent edits. Skill bundle: skill-manifest.json.