How to find the Direct Discrete Laplace Transform of ( 2 n n )

hemotropS7A

hemotropS7A

Answered

2022-11-24

How to find the Direct Discrete Laplace Transform of ( 2 n n )

Answer & Explanation

Kenyon Valencia

Kenyon Valencia

Expert

2022-11-25Added 11 answers

The generating function of Catalan's numbers y n = 1 n + 1 ( 2 n n ) is
n 0 1 n + 1 ( 2 n n ) x n = 1 1 4 x 2 x
Multplying by x and differentitating gives that
n 0 ( 2 n n ) x n = d d x 1 1 4 x 2 = 1 1 4 x
One can also try to prove this directly, by noting that
( 1 / 2 k ) = ( 1 ) k 4 k ( 2 k k )
and that
( 1 + x ) α = n 0 ( α n ) x n
At any rate, any claim begs for a proof.

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