Bell Triangle Entry

Return the zero-indexed Aitken Bell triangle entry with row zero [1], each row starting at the previous row’s last entry, and a 300-row resource limit.

Description

The Aitken Bell triangle builds set-partition counts by successive row additions.

The Aitken Bell triangle builds set-partition counts by successive row additions. Its leftmost entries are Bell numbers under zero-based row indexing.^[1]

The tool returns value. The indexing and output conventions below determine how to read these values.

When to use Bell Triangle Entry

Use this for the following task: Return the zero-indexed Aitken Bell triangle entry with row zero [1], each row starting at the previous row’s last entry, and a 300-row resource limit. Use the input and output conventions below to reproduce the intended mathematical quantity.

How to use Bell Triangle Entry and understand its results

Choose the required indexes, integers or sequence entries. Preserve decimal-string encodings where the field table specifies strings.
Enter n, k according to the field table, then run the tool.
Read value. Preserve exact decimal strings and the stated indexing or coefficient order.
Keep exact decimal strings for subsequent integer arithmetic; distinguish them from any explicitly approximate numeric result.

Input field	Meaning	Value type and bounds	Requirement
n	Zero-indexed row (resource limit 300)	integer; minimum 0; maximum 300	Required
k	Zero-indexed column (resource limit 300)	integer; minimum 0; maximum 300	Required

Build BigInt rows through n and return column k as decimal text. Require k≤n in addition to the declared index bounds.

row(0)=[1]; row(i)[0]=row(i−1).last; row(i)[j]=row(i)[j−1]+row(i−1)[j−1]

The symbols and boundary conditions refer to the supplied fields and the implementation described above.

Output field	Meaning	Value type
value	Aitken Bell triangle entry	string (integer/arbitrary)

json

{
  "input": {
    "n": 3,
    "k": 2
  },
  "output": {
    "value": "10"
  }
}

Limitations and assumptions

Rows and columns are zero-based and at most 300. An out-of-triangle column is an error rather than a zero entry.

Use the declared integer bounds and string formats. Large exact outputs remain decimal strings; numerical approximations use ordinary floating-point arithmetic. A rejected input is different from a valid zero count or false membership flag.

Alternative or Complementary methods

Choose Bell Number when you need its operation: Count partitions of an n-element labeled set as an exact decimal integer; n is capped at 300 for bounded computation. Choose Pascal Triangle Row for the complementary operation: Return the zero-indexed nth row of Pascal’s triangle as exact decimal binomial coefficients; n is capped at 1,000 to bound the output.

Background reference: OEIS A011971.

References

OEIS A011971 — OEIS Foundation

Similar or alternative tools

Bell Number
Count partitions of an n-element labeled set as an exact decimal integer; n is capped at 300 for bounded computation.
Pascal Triangle Row
Return the zero-indexed nth row of Pascal’s triangle as exact decimal binomial coefficients; n is capped at 1,000 to bound the output.