Lah Transform Calculator
Compute exact unsigned Lah transform b_n=Σ T(n,k)a_k, preserving integer output as strings and index-zero convention; bounded to n≤64, not a named-number lookup.
Description
Unsigned Lah numbers count partitions into ordered lists and define a sequence transform.
Unsigned Lah numbers count partitions into ordered lists and define a sequence transform. Input starts at index zero, which must be preserved in the output.1
Compute exact unsigned Lah transform b_n=Σ T(n,k)a_k, preserving integer output as strings and index-zero convention; bounded to n≤64, not a named-number lookup.
When to use Lah Transform Calculator
Use this tool to compute exact unsigned Lah transform b_n=Σ T(n,k)a_k, preserving integer output as strings and index-zero convention; bounded to n≤64, not a named-number lookup.
Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
How to use Lah Transform Calculator and understand its results
- Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
- Enter integerSequenceFromIndexZero using the field table and calculation rules, then run the tool.
- Read exactTransformedSequence. The explanation below defines the units, signs, ordering and any flags or placeholder values.
- If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.
| Input field | Meaning | Value type; see rules below | Requirement |
|---|---|---|---|
| integerSequenceFromIndexZero | Integer sequence from index zero | list of number | Required |
Supply 1–65 signed integer coefficients starting a0, each |value|≤one million. Accumulate exact BigInt Lah weights and return decimal strings for every transformed index.
b_n = Σ[k=1..n]C(n−1,k−1)n!/k! × a_k; b0=a0 Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.
| Output field | Meaning | Value type |
|---|---|---|
| exactTransformedSequence | Exact transformed sequence | list of string |
{
"input": {
"integerSequenceFromIndexZero": [
0,
1,
1,
1,
1
]
},
"output": {
"exactTransformedSequence": [
"0",
"1",
"3",
"13",
"73"
]
}
}Limitations and assumptions
Only the finite supplied sequence is transformed. Numeric parsing can lose exact large outputs.
Numeric inputs and returned numeric values must be finite; an unrepresentable returned value produces an error. Numeric lists are limited to 100,000 entries; the rules above may require smaller limits. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.
Alternative or Complementary methods
Use Stirling Transform Calculator: Compute exact second-kind Stirling transform b_n=Σ T(n,k)a_k, preserving integer output as strings and index-zero convention; bounded to n≤64, not a named-number lookup. For a complementary task, use Bell Transform Calculator: Compute complete exponential Bell-polynomial transform with b0=1 and b_n=Σ C(n−1,k−1)a_k b_(n−k); caller sequence differs materially from a single Bell number.
References
-
Lah Transform Calculator — tool.io
Similar or alternative tools
- Stirling Transform Calculator
Compute exact second-kind Stirling transform b_n=Σ T(n,k)a_k, preserving integer output as strings and index-zero convention; bounded to n≤64, not a named-number lookup.
- Bell Transform Calculator
Compute complete exponential Bell-polynomial transform with b0=1 and b_n=Σ C(n−1,k−1)a_k b_(n−k); caller sequence differs materially from a single Bell number.