Does the description disclose side effects, auth requirements, rate limits, or destructive behavior?

Beyond the annotations (readOnly, idempotent, openWorld), the description adds pricing ($0.001 USDC per successful call via x402 on Solana) and an HTTPS endpoint alternative, which are genuinely useful operational facts. The edge-case rule for k > n/2 also clarifies behavior; only the exact return type/value shape is left implicit.

Agents need to know what a tool does to the world before calling it. Descriptions should go beyond structured annotations to explain consequences.

Is the description appropriately sized, front-loaded, and free of redundancy?

Front-loaded with the mathematical definition, then example, boundary rule, ranges, and finally cost/endpoint. Nearly every sentence carries information; the pricing and endpoint sentences are dense but each serves a distinct operational need.

Shorter descriptions cost fewer tokens and are easier for agents to parse. Every sentence should earn its place.

Given the tool's complexity, does the description cover enough for an agent to succeed on first attempt?

For a 2-parameter read-only math tool with no output schema, the description supplies the formula, a worked example, and the boundary behavior, which is enough to call it correctly and predict the result. The only gap is not stating the return type explicitly, though the example implies an integer.

Complex tools with many parameters or behaviors need more documentation. Simple tools need less. This dimension scales expectations accordingly.

Does the description clarify parameter syntax, constraints, interactions, or defaults beyond what the schema provides?

With 0% schema description coverage, the description must carry the parameters, and it does: n is bounded 0–30, k is bounded 0–15, k is the index selecting which summand, and k past n/2 yields 0. It adds real semantic meaning (k as an index, the zero boundary) beyond the bare min/max in the schema.

Input schemas describe structure but not intent. Descriptions should explain non-obvious parameter relationships and valid value ranges.

Does the description clearly state what the tool does and how it differs from similar tools?

The description gives the exact formula C(n,2k)*C(2k,k) for a single summand of the central trinomial coefficient, which is a precise verb+resource statement. It implicitly distinguishes itself from aggregation siblings (trinomial_sum, trinomial) by saying 'summand', though it never names those alternatives explicitly.

Agents choose between tools based on descriptions. A clear purpose with a specific verb and resource helps agents select the right tool.

Does the description explain when to use this tool, when not to, or what alternatives exist?

Usage context is conveyed through the n=5 example and the boundary rule that k past n/2 returns 0, which tells the agent what inputs are meaningful. However, there is no explicit guidance on when to choose this tool over trinomial_sum or trinomial, so the agent must infer the distinction.

Agents often have multiple tools that could apply. Explicit usage guidance like "use X instead of Y when Z" prevents misuse.