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.

Description

The inverse binomial transform recovers a sequence prefix by an alternating binomial sum.

The inverse binomial transform recovers a sequence prefix by an alternating binomial sum. It reverses the ordinary zero-indexed binomial transform.^[1]

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

When to use Inverse Binomial Transform

Use this for the following task: 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. Use the input and output conventions below to reproduce the intended mathematical quantity.

How to use Inverse Binomial Transform 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	Binomial transform 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 integer strings with BigInt, build each row’s binomial coefficients incrementally and add or subtract weighted terms according to parity.

a(n)=Σ[k=0..n](−1)^(n−k)C(n,k)b(k)

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

Output field	Meaning	Value type
values	Recovered sequence prefix	list of string (integer/arbitrary)

json

{
  "input": {
    "values": [
      "1",
      "3",
      "8",
      "20"
    ]
  },
  "output": {
    "values": [
      "1",
      "2",
      "3",
      "4"
    ]
  }
}

Limitations and assumptions

The first entry represents index zero. Empty lists are allowed and prefixes are capped at 256 terms; the transformation is not a divisor-sum Möbius inversion.

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 Binomial Transform of an Integer Sequence when you need its operation: 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. Choose Möbius Inversion of Divisor Sums for the complementary operation: Recover f(1)…f(N) from F(n)=Σ(d|n)f(d) by exact triangular divisor-sum inversion; prefixes start at one and are capped at 1,000 entries.

Background reference: Binomial transform.

References

Binomial transform — OEIS Foundation

Similar or alternative tools

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.
Möbius Inversion of Divisor Sums
Recover f(1)…f(N) from F(n)=Σ(d|n)f(d) by exact triangular divisor-sum inversion; prefixes start at one and are capped at 1,000 entries.