Talk:Trinomial triangle
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Central Trinomial Coefficient Formula
I got the formula for central Trinomial coefficients: (n 0)2 = Σ nCk * kC(n - k) from k = n/2 if n is even or k = (n + 1)/2 if n is odd till k = n.
If you select n numbers from 2n numbers with each number having its duplicate, first you select k different numbers from n numbers and select n - k other numbers which will be duplicates of few of the k different numbers. Then you will select n cards from 2n cards. If you select n/2 cards for n even, you will select n - n/2 = n/2 cards. n/2 is the least number which has to be added to itself or smaller number to equal n. Similarly, (n + 1)/2 is the least number which has to be added to itself or smaller number to equal n. Here, n/2 and (n + 1)/2 are natural numbers.
The general result also goes by the same logic and is: (n r)2 = Σ nCk * kC(m - k) where r = m - n and k is from m/2 if m is even or k is from (m + 1)/2 if m is odd till k = m. 2409:40E0:F5:2AD1:94EE:2EF0:445C:6B77 (talk) 13:24, 22 October 2024 (UTC)