Pascal's Triangle Row Calculator
Enter a row index n and instantly get the full nth row of Pascal's triangle along with its sum and largest coefficient.
Sum of all coefficients in this row
- 1
C(n,0) = 1
1Every row starts with 1. - 2
C(n,1) = C(n,0) × n ÷ 1
1 × 6 ÷ 1 = 6 - 3
Row sum = 2^n
2^6 = 64The sum of all binomial coefficients in row n equals 2^n (binomial theorem with x=y=1).
How does this calculator work?
Enter a row index n and this tool returns the nth row of Pascal's triangle—the binomial coefficients C(n,0) through C(n,n). It builds them with the recurrence C(n,k) = C(n,k−1)·(n−k+1)/k, shows the symmetric bell shape, the number of terms (n+1), and confirms the row sum equals 2^n.
Formula
How this is calculated
Pascal's triangle is the array of binomial coefficients, where each entry equals the sum of the two entries directly above it. The single input n is the 0-indexed row number: row 0 is [1], row 1 is [1, 1], row 2 is [1, 2, 1], and so on. The calculator builds the row iteratively rather than computing each factorial separately. It starts with C(n,0) = 1 and applies the recurrence C(n,k) = C(n,k−1) · (n−k+1) / k for k = 1 up to n. Because every intermediate value is an exact integer, this multiplicative method avoids the overflow you would get from computing n! directly and is far faster.
The row has n + 1 terms, is symmetric (C(n,k) = C(n,n−k)), and forms a bell-shaped distribution peaking in the middle. The sum of all coefficients in row n equals 2^n, which follows from the binomial theorem with x = y = 1: (1 + 1)^n = Σ C(n,k). These coefficients are exactly the multipliers that appear when you expand (x + y)^n.
The input must be a non-negative integer. Non-integer, negative, or empty values produce no result. For very large n the coefficients exceed the precision of double-precision floating point, so results above a few hundred should be treated as approximate.
Frequently asked questions
Row 0 is simply [1]. The triangle is 0-indexed, so the topmost single 1 is row 0, [1, 1] is row 1, and [1, 2, 1] is row 2.
Setting x = y = 1 in the binomial theorem gives (1 + 1)^n = Σ C(n,k), so the coefficients of row n always add up to 2^n.
The calculator uses the recurrence C(n,k) = C(n,k−1)·(n−k+1)/k starting from C(n,0) = 1, which keeps every step an exact integer and avoids factorial overflow.
Also known as
TG we-Calculate Editorial Team. (2026). Pascal's Triangle Row Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/pascals-triangle-row-calculator
TG we-Calculate Editorial Team. "Pascal's Triangle Row Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/pascals-triangle-row-calculator.
TG we-Calculate Editorial Team, "Pascal's Triangle Row Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/pascals-triangle-row-calculator
@misc{wecalculate_pascals_triangle_row_calculator, title = {Pascal's Triangle Row Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/pascals-triangle-row-calculator}}, year = {2026}, note = {TG we-Calculate} }
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