Binomial Transform of an Integer Sequence
Transform a zero-indexed prefix by b(n)=ΣC(n,k)a(k) using exact decimal integers; empty prefixes are allowed and input is capped at 256 terms.
Description
The binomial transform weights each prefix entry by the binomial coefficients of the output index.
The binomial transform weights each prefix entry by the binomial coefficients of the output index. Its sequence indexing begins at zero.1
The tool returns values. The indexing and output conventions below determine how to read these values.
When to use Binomial Transform of an Integer Sequence
Use this for the following task: Transform a zero-indexed prefix by b(n)=ΣC(n,k)a(k) using exact decimal integers; empty prefixes are allowed and input is capped at 256 terms. Use the input and output conventions below to reproduce the intended mathematical quantity.
How to use Binomial Transform of an Integer Sequence and understand its results
- Choose the required indexes, integers or sequence entries. Preserve decimal-string encodings where the field table specifies strings.
- Enter values according to the field table, then run the tool.
- Read values. 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 |
|---|---|---|---|
| values | Sequence prefix starting at index zero | list of string (integer/arbitrary); minimum entries 0; maximum entries 256; each entry: maximum text length 200; pattern ^-?(0|[1-9][0-9]*)$ | Required |
Parse signed decimal integers with BigInt, generate each row of binomial coefficients incrementally and accumulate exact weighted sums.
b(n)=Σ[k=0..n]C(n,k)a(k) The symbols and boundary conditions refer to the supplied fields and the implementation described above.
| Output field | Meaning | Value type |
|---|---|---|
| values | Binomial transform prefix | list of string (integer/arbitrary) |
{
"input": {
"values": [
"1",
"2",
"3",
"4"
]
},
"output": {
"values": [
"1",
"3",
"8",
"20"
]
}
}Limitations and assumptions
Empty prefixes are allowed and at most 256 terms are accepted. The tool does not infer values beyond the prefix; use the inverse transform to reverse this operation.
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 Inverse Binomial Transform when you need its operation: Invert a zero-indexed binomial-transform prefix by a(n)=Σ(−1)^(n−k)C(n,k)b(k) using exact decimal integers and at most 256 terms. Choose Dirichlet Convolution Prefix for the complementary operation: Compute h(n)=Σ(d|n)f(d)g(n/d) for equally sized arithmetic-function prefixes starting at n=1, using exact integers and at most 1,000 terms.
Background reference: Binomial transform.
References
-
Binomial transform — OEIS Foundation
Similar or alternative tools
- Inverse Binomial Transform
Invert a zero-indexed binomial-transform prefix by a(n)=Σ(−1)^(n−k)C(n,k)b(k) using exact decimal integers and at most 256 terms.
- Dirichlet Convolution Prefix
Compute h(n)=Σ(d|n)f(d)g(n/d) for equally sized arithmetic-function prefixes starting at n=1, using exact integers and at most 1,000 terms.