Wikipedia:Reference desk/Archives/Mathematics/2023 April 6

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

April 6

30th row of Pascal's triangle

The first row is simply 1. The second row is 1-1. The third row is 1-2-1. The fourth row is 1-3-3-1. The fifth row is 1-4-6-4-1. And so on. Do you know the 30th row?? (It's easy to guess correctly that the first 2 points in it are 1-29 and the last 2 are 29-1, but do you know all 30 members of the 30th row of Pascal's triangle?? Georgia guy (talk) 14:33, 6 April 2023 (UTC)

As described at Pascal's triangle, it's based on a simple, easily repeatable algorithm, and there are several options to find it. You can do it by brute force by simply repeating the algorithm yourself, you can apply the process described in that article in the section titled "Calculating a row or diagonal by itself". Or you can use an online calculator such as this one, which will generate any row you wish. --Jayron32 14:43, 6 April 2023 (UTC)

(ec) Sure, it's $30 \choose 0$, $30 \choose 1$, ..., $30 \choose 30$, cf. binomial coefficient (or simply go through the triangle until you reach row 30). As I do not have more time than you have, I won't do the numbers for you. --Wrongfilter (talk) 14:49, 6 April 2023 (UTC)

The 30th row has big numbers; the largest is more than 70 million. (The 15th row is 1-14-91-364-1001-2002-3003-3432-3003-2002-1001-364-91-14-1.) Georgia guy (talk) 14:52, 6 April 2023 (UTC)

Georgia guy: If you're already aware of how to find the 30th row, and are aware of it enough to know the scale of the numbers on it, what kind of answer were you seeking? --Jayron32 14:56, 6 April 2023 (UTC)

User:Jayron32, the main thing I want to know is what row of Pascal's triangle has "2n, 3n, 4n" as 3 consecutive terms. The 15th row has n, 2n, and 3n. (To find what row of the triangle you're on, just add one to the second number in the row; the 30th row starts with 1,29... Georgia guy (talk) 15:38, 6 April 2023 (UTC)

Well then you should have asked exactly that. If you just wanted to know the 30th row, that's academic. If you want to know which rows have some specific property that's a different question. The 30th row doesn't have that property. It has an "n 2n" sequence (10015005, 20030010) but the pattern breaks on either side of it. In researching your question, I also found that, in Commons, the image File:Pascals triangle 30 lines.png exists. You can explore it yourself. --Jayron32 16:15, 6 April 2023 (UTC)

I see that my guess of the 30th row is wrong. The rows where 3n and 4n occur as consecutive members of the triangle are the rows divisible by 7, which 30 is not. So the row that would have 2n,3n,4n is not the 30th, but an even lower row, the 35th. Its terms are 1,34,561...561,34,1. Georgia guy (talk) 16:20, 6 April 2023 (UTC)

OK. --Jayron32 16:36, 6 April 2023 (UTC)

Yes, it is the 35th row (the row that is designated as "34" by people who use cardinal numbers to designate rows of Pascal's triangle.) The value of n where 2n, 3n, and 4n occur on this row is 463,991,880. Georgia guy (talk) 16:53, 6 April 2023 (UTC)

OK. --Jayron32 17:43, 6 April 2023 (UTC)

Enter for(n=0,30,print1(binomial(30,n)", ")) at https://pari.math.u-bordeaux.fr/gp.html. vector(31,n,binomial(30,n-1)) is shorter but may be less instructive if you want to do other problems. You can also download PARI/GP and run it on your own computer. PrimeHunter (talk) 15:05, 6 April 2023 (UTC)

The 3435th row is the one you're looking for, elements 13, 14 and 15, with n = 463,991,880 (if I'm not mistaken). It's fairly easy to find, the conditions are ${N \choose k+1}={\frac {3}{2}}{N \choose k}$ and ${N \choose k+2}={\frac {4}{3}}{N \choose k+1}$. A lot of terms cancel and these conditions lead to two linear equations with solution N=34 and k=13. (correction: the triangle starts with N=0, so N=34 is the 35th row as you would have it). --Wrongfilter (talk) 16:35, 6 April 2023 (UTC)

In general, the system of equations

${\binom {N}{k}}:{\binom {N}{k{+}1}}:{\binom {N}{k{+}2}}=p:p{+}1:p{+}2$

is solved by

$N=4p^{2}+8p+2,k=2p^{2}+3p-1.$

--Lambiam 21:20, 6 April 2023 (UTC); corrected 06:48, 7 April 2023 (UTC)

If you only require ${\binom {N}{k}},{\binom {N}{k{+}1}},{\binom {N}{k{+}2}}$ to be in arithmetic sequence then you get these solutions, those obtained by the symmetry ${\binom {N}{k}}={\binom {N}{N-k}},$ and no others. --RDBury (talk) 12:58, 9 April 2023 (UTC)

You already said that the first number in the row is binomial(29, 0) = 1. You can compute each entry in the row from the previous one by binomial(29, k) = ((29 + 1 - k) / k) · binomial(k - 1). You get:

1, 29, 406, 3654, 23751, 118755, 475020, 1560780, 4292145, 10015005, 20030010, 34597290, 51895935, 67863915, 77558760, 77558760, 67863915, 51895935, 34597290, 20030010, 10015005, 4292145, 1560780, 475020, 118755, 23751, 3654, 406, 29, 1

– b_jonas 15:33, 7 April 2023 (UTC)