PASCAL

Constructs an n-by-n Pascal matrix. A Pascal matrix contains the binomial coefficients as its entries.

Excel Usage

=PASCAL(n, pascal_kind)

n (int, required): The order of the matrix.
pascal_kind (str, optional, default: “symmetric”): Type of Pascal matrix to return (symmetric, lower, or upper).

Returns (list[list]): 2D array representing the Pascal matrix.

Example 1: Symmetric Pascal matrix of order 3

Inputs:

n	pascal_kind
3	symmetric

Excel formula:

=PASCAL(3, "symmetric")

Expected output:

Result
1	1	1
1	2	3
1	3	6

Example 2: Lower triangular Pascal matrix

Inputs:

n	pascal_kind
3	lower

Excel formula:

=PASCAL(3, "lower")

Expected output:

Result
1	0	0
1	1	0
1	2	1

Example 3: Upper triangular Pascal matrix

Inputs:

n	pascal_kind
4	upper

Excel formula:

=PASCAL(4, "upper")

Expected output:

Result
1	1	1	1
0	1	2	3
0	0	1	3
0	0	0	1

Example 4: Pascal matrix with single-cell range order

Inputs:

n	pascal_kind
5	symmetric

Excel formula:

=PASCAL({5}, "symmetric")

Expected output:

Result
1	1	1	1	1
1	2	3	4	5
1	3	6	10	15
1	4	10	20	35
1	5	15	35	70

Python Code

import numpy as np
from scipy.linalg import pascal as scipy_pascal

def pascal(n, pascal_kind='symmetric'):
    """
    Construct a Pascal matrix.

    See: https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.pascal.html

    This example function is provided as-is without any representation of accuracy.

    Args:
        n (int): The order of the matrix.
        pascal_kind (str, optional): Type of Pascal matrix to return (symmetric, lower, or upper). Valid options: Symmetric, Lower Triangular, Upper Triangular. Default is 'symmetric'.

    Returns:
        list[list]: 2D array representing the Pascal matrix.
    """
    try:
        if isinstance(n, list):
            if not n or not isinstance(n[0], list) or not n[0]:
                return "Error: n must be a scalar or single-cell range"
            n = n[0][0]

        try:
            n_val = int(n)
        except (TypeError, ValueError):
            return "Error: n must be an integer"

        if n_val <= 0:
            return "Error: n must be a positive integer"

        if not isinstance(pascal_kind, str):
            return "Error: pascal_kind must be a string"

        kind = pascal_kind.lower()
        if kind not in ["symmetric", "lower", "upper"]:
            return "Error: pascal_kind must be one of symmetric, lower, or upper"

        res = scipy_pascal(n_val, kind=kind)
        return res.tolist()

    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

n *

The order of the matrix.

pascal_kind

Type of Pascal matrix to return (symmetric, lower, or upper).