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.
Description
The complete exponential Bell transform maps a sequence of exponential-series coefficients to a new coefficient sequence.
The complete exponential Bell transform maps a sequence of exponential-series coefficients to a new coefficient sequence. It includes an initial output b0=1 independent of input starting at a1.1
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.
When to use Bell Transform Calculator
Use this tool to 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.
Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
How to use Bell 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 exponentialSeriesCoefficientsFromIndexOne using the field table and calculation rules, then run the tool.
- Read completeBellTransform. 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 |
|---|---|---|---|
| exponentialSeriesCoefficientsFromIndexOne | Exponential series coefficients from index one | list of number | Required |
Supply 1–64 signed integer coefficients a1 through aN, each magnitude ≤one million. Accumulate the recurrence with exact BigInt arithmetic and return decimal strings for b0 through bN.
b0=1; b_n=Σ[k=1..n]C(n−1,k−1)a_k b_(n−k) Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.
| Output field | Meaning | Value type |
|---|---|---|
| completeBellTransform | Complete bell transform | list of string |
{
"input": {
"exponentialSeriesCoefficientsFromIndexOne": [
1,
1,
1,
1
]
},
"output": {
"completeBellTransform": [
"1",
"1",
"2",
"5",
"15"
]
}
}Limitations and assumptions
This is a sequence transform rather than one Bell-number lookup. Preserve exact strings for large output.
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 Partition Transform Calculator: Compute exact coefficients through N of formal product ∏(1−x^k)^(-a_k), allowing signed caller exponent sequence; not ordinary unrestricted partition count or finite q product evaluation.
References
-
Bell 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.
- Partition Transform Calculator
Compute exact coefficients through N of formal product ∏(1−x^k)^(-a_k), allowing signed caller exponent sequence; not ordinary unrestricted partition count or finite q product evaluation.