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CoefficientList

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

CoefficientList[poly, var] gives a list of coefficients of powers of var in poly, starting with power 0.
CoefficientList[poly, {var1, var2, ...}] gives an array of coefficients of the variables.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= CoefficientList[1 + 6 x - x^4, x]
Out[1]= {1, 6, 0, 0, -1}

In[2]:= CoefficientList[(1 + x)^10, x]
Out[2]= {1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1}

In[3]:= CoefficientList[1 + a x^2 + b x y + c y^2, {x, y}]
Out[3]= {{1, 0, c}, {0, b, 0}, {a, 0, 0}}

Implementation notes

Algorithm. builtin_coefficientlist (in src/poly/poly.c) returns the dense coefficient array of a polynomial. It expr_expands the input, computes each variable's degree with get_degree_poly, then the recursive worker coeff_list_rec builds a nested List whose shape mirrors the variable order. At each level it pulls all coefficients c_0..c_d of the current variable — preferring the bulk extractor get_all_coeffs_expanded (single pass over the expanded form) and falling back to get_coeff_expanded per degree — and recurses on each coefficient for the next variable.

Data structures. A int* max_degrees array sizes each axis; coefficients are produced as Expr* and assembled into nested List nodes. The bulk path avoids the naive (degree+1)-passes-per-level cost.

  • Protected.
  • Gives an array of coefficients starting with power 0.
  • Returns a full rectangular array for multiple variables. Combinations of powers that do not appear in poly give zeros in the array.
  • Automatically expands the polynomial internally.

Attributes: Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= CoefficientList[x^2 + 3 x + 2, x]
Out[1]= {2, 3, 1}

In[1]:= CoefficientList[1 + x^3, x]
Out[1]= {1, 0, 0, 1}

In[1]:= CoefficientList[a + b x + c x^2, x]
Out[1]= {a, b, c}

In[1]:= CoefficientList[x^2 + x y + y^2, {x, y}]
Out[1]= {{0, 0, 1}, {0, 1, 0}, {1, 0, 0}}

In[1]:= CoefficientList[(1 + x)^6, x]
Out[1]= {1, 6, 15, 20, 15, 6, 1}

In[1]:= CoefficientList[Expand[(1 + x + x^2)^4], x]
Out[1]= {1, 4, 10, 16, 19, 16, 10, 4, 1}

Notes

CoefficientList[poly, var] returns the dense coefficient vector ordered by ascending power, starting at the constant term (power 0); missing powers are filled with explicit zeros, as the {1, 0, 0, 1} for 1 + x^3 shows. The list length is one more than the degree. With a list of variables, the result is a nested array whose [[i, j]] entry is the coefficient of x^(i-1) y^(j-1), so x^2 + x y + y^2 produces a 3x3 matrix with ones on the anti-diagonal. Coefficients may be symbolic, preserving parameters such as a, b, c. Applied to (1 + x)^6 it reads off a row of Pascal's triangle ({1, 6, 15, 20, 15, 6, 1}), and the (pre-expanded) trinomial (1 + x + x^2)^4 yields the symmetric central-trinomial coefficient row {1, 4, 10, 16, 19, 16, 10, 4, 1} — a quick way to read off such combinatorial sequences.