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

Annotations already cover the safety profile (read-only, idempotent, non-destructive, open-world). The description adds useful behavioral context beyond the annotations: the accepted n range, the output formatting rule past 15 digits, the per-call price, and an alternate HTTPS endpoint.

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?

The definition is front-loaded with the mathematical identity and examples, then follows with range, output formatting, and payment/endpoint details. It is compact and most sentences earn their place, though the endpoint sentence is secondary.

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 simple one-parameter math tool with no output schema, the description supplies the range, examples, and an output-format caveat. It is nearly complete, though it could explicitly state the normal return type for values up to 15 digits.

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?

Schema description coverage is 0%, so the description must carry the burden for the single parameter n. It explains that n runs from 0 to 64 and gives example values, but because the schema already enforces the same min/max, the added semantic value is moderate rather than complete.

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 names a specific mathematical object, the central Delannoy number D(n,n), which distinguishes it from the sibling tools delannoy and delannoy_sum by defining the central diagonal case. It is clear but does not explicitly route the agent away from those siblings.

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?

The description states what the tool computes but gives no when-to-use guidance, no exclusions, and no explicit alternatives such as delannoy for non-central values or delannoy_sum for summations.

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