Trinomial triangle (Python)

Question

As part of my interest in learning Python, I've hit a stop when coming across an exercise that states:

Consider the expression (1 + x + x^2)^n and write a program which calculates a modified Pascal’s triangle (known as the trinomial triangle) for the coefficients of its expansion. Can you come up with a simple rule (and prove that it works!) which would enable you to write down the coefficients of this triangle?

So, I'm trying to write a code that prints out the trinomial triangle from a user input. This is the code I have so far:

import math

rows = 0 #We count how many rows we print

#We define a function that will calculate the triangle.
def calc(n, r):
    if r == 0 or r == n:
        return 1
    return int(math.pow(1 + r + math.pow(r, 2), n))

#We define a function that append the values in an array.
def triangle(rows):
    result = [] #We need an array to collect our results.
    for count in range(rows): #For each count in the inputted rows
        row = [] #We need a row element to collect the rows.
        for element in range(count + 1):
            #We add the function calculation to the row array.
            row.append(calc(count, element))
        #We add the row(s) to the result array.
        result.append(row)
    return result

number = int(input("How many rows do you want printed? "))

#We can now print the results:
for row in triangle(number):
    rows += 1 #We add one count to the amount of rows
    print("Row %d: %s" % (rows, row)) #Print everything

which returns

How many rows do you want printed? 5
Row 1: [1]
Row 2: [1, 1]
Row 3: [1, 9, 1]
Row 4: [1, 27, 343, 1]
Row 5: [1, 81, 2401, 28561, 1]

And as I understand it, the expected result should be:

1
1   1   1
1   2   3   2   1
1   3   6   7   6   3   1
1   4   10  16  19  16  10  4  1

I don't exactly know how to proceed from here. Any suggestions to point me in the right direction would be appreciated.

Why not use combination formulas? See https://en.wikipedia.org/wiki/Pascal%27s_triangle — pjpj, Oct 01 '17 at 16:04
Because then I'm left with the "original" Pascal's triangle. I was trying to implement the (1 + x + x^2)^n in my code instead of factorizing in the calculation. — tsvenbla, Oct 01 '17 at 16:08
But your expected result is the original one... what do you mean? — pjpj, Oct 01 '17 at 16:16
Do you know the rule for computing a row from the previous row in the standard binomial Pascal's Triangle? There's a very similar rule for this trinomial triangle. — PM 2Ring, Oct 01 '17 at 16:16
@WendingPeng No, that expected result is _not_ the normal binomial Pascal's Triangle. Look closer! — PM 2Ring, Oct 01 '17 at 16:17
So have you tried http://mathworld.wolfram.com/TrinomialCoefficient.html ? — pjpj, Oct 01 '17 at 16:21

score 1 · Accepted Answer · answered Oct 01 '17 at 16:21

In the usual binomial version of Pascal's Triangle we can compute each value in a row by adding the two entries immediately above it. In the trinomial version we add three entries. The proof of this isn't hard, so I'll let you figure it out. ;)

Here's one way to do that in Python.

row = [1]
for i in range(8):
    print(row)
    row = [sum(t) for t in zip([0,0]+row, [0]+row+[0], row+[0,0])]

output

[1]
[1, 1, 1]
[1, 2, 3, 2, 1]
[1, 3, 6, 7, 6, 3, 1]
[1, 4, 10, 16, 19, 16, 10, 4, 1]
[1, 5, 15, 30, 45, 51, 45, 30, 15, 5, 1]
[1, 6, 21, 50, 90, 126, 141, 126, 90, 50, 21, 6, 1]
[1, 7, 28, 77, 161, 266, 357, 393, 357, 266, 161, 77, 28, 7, 1]

Hm, I never looked at it like that before I guess. I just used the calculation part to compute the binomial Pascal's Triangle. Thanks for letting me know about the "add entities directly above"! Kudos. — tsvenbla, Oct 01 '17 at 16:24

score -1 · Answer 2 · answered Jul 11 '18 at 02:38

-1

And just so that the line that I want to show appears as for example income 2 only shows me the second line of the triangle that is 1 1 1

answered Jul 11 '18 at 02:38

Reinaldo

1

Trinomial triangle (Python)

2 Answers2