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.
Description
The second-kind Stirling transform converts an index-zero sequence through Stirling-number weights.
The second-kind Stirling transform converts an index-zero sequence through Stirling-number weights. Exact integer outputs can exceed JavaScript's safe numeric range.1
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.
When to use Stirling Transform Calculator
Use this tool to 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.
Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
How to use Stirling 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 a0 onward, each with absolute value at most one million. Build Stirling rows and accumulate with BigInt. Return decimal strings for all transformed entries, preserving b0=a0.
b_n = Σ[k=0..n]S(n,k)a_k; S(n,k)=S(n−1,k−1)+kS(n−1,k) 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",
"2",
"5",
"15"
]
}
}Limitations and assumptions
The finite supplied sequence defines the output horizon. Parse output strings as exact integers if needed, rather than ordinary floating-point numbers.
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 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. 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
-
Stirling Transform Calculator — tool.io
Similar or alternative tools
- 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.
- 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.