Find the generating function of f ( n ) = <munderover> &#x2211;<!-- ∑ --> <mrow

sebadillab0 2022-07-08 Answered
Find the generating function of f ( n ) = k = 0 n ( n k ) ( 1 ) n k C k
I want to find the generating function of f ( n ) = k = 0 n ( n k ) ( 1 ) n k C k , where C k is the k-th Catalan number. So, using the definition of an ordinary generating function:
F ( x ) = n 0 ( k = 0 n ( n k ) ( 1 ) n k C k ) x n
recalling that: C ( x ) = n 0 C n x n = 1 1 4 x 2 x
The first idea was to see F(x) as a product of two formal series and I have already seen a proof in this regard where they use the theorem of residues, yet I am looking for something less refined. Any idea?
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Answers (1)

Alexia Hart
Answered 2022-07-09 Author has 19 answers
Explanation:
F ( x ) = n 0 ( k = 0 n ( n k ) ( 1 ) n k C k ) x n = k 0 ( 1 ) k C k n k ( n k ) ( x ) n = k 0 ( 1 ) k C k ( x ) k ( 1 + x ) k + 1 = 1 1 + x k 0 C k ( x 1 + x ) k = 1 1 + x 1 1 4 x 1 + x 2 x 1 + x = 1 1 3 x 1 + x 2 x
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1. There is a real numbers x that for all real number y, 2 x + y = 7 will be true.
would this be false because if you say x = 6
then you get
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only if y = 2 would it work but it would not work for every y.

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