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?

Question

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&#039;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?

Royce Golden · Accepted Answer

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      !

Landen Miller · Accepted Answer

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            !

Factorial equality (135*(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?

Open question

Answer & Explanation

New Questions in Pre-Algebra