Factorial equality (1*3*5*(2k-1))/(2^k2!}=((2k)!)/(2^k2^kk!k!) In a generating function identity proof in my textbook there is a step that I can't wrap my head around. How does one get from the left side of the equation to the right side? Is there an intuitive explanation as for why this makes sense?

Samson Kaufman 2022-08-16 Answered
Factorial equality   1 3 5 ( 2 k 1 ) 2 k 2 !   = ( 2 k ) ! 2 k 2 k k ! k !
In a generating function identity proof in my textbook there is a step that I can't wrap my head around.
1 3 5 ( 2 k 1 ) 2 k 2 !
= ( 2 k ) ! 2 k 2 k k ! k !
How does one get from the left side of the equation to the right side? Is there an intuitive explanation as for why this makes sense?
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Answers (2)

Royce Golden
Answered 2022-08-17 Author has 12 answers
HINT:
Note that we can write
1 3 5 ( 2 k 1 ) = 1 2 3 4 ( 2 k 2 ) ( 2 k 1 ) ( 2 k ) 2 ( 1 ) 2 ( 2 ) 2 ( k 1 ) 2 ( k ) = ( 2 k ) ! 2 k k !
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Landen Miller
Answered 2022-08-18 Author has 4 answers
1 3 5 7 11 13 2 7 2 ! = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 2 7 2 4 6 8 10 12 14 = 14 ! 2 7 ( 2 2 2 2 2 2 2 ) ( 1 2 3 4 5 6 7 ) = 14 ! 2 7 2 7 7 !
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