Expression for binomial coefficient denominatorI'm trying to find an analytical expression for the denominator of ( − 1 / 2 k ) in terms of k when the fraction is fully reduced.E.g., the first several such denominators, starting with k = 0, are 1 , 2 , 8 , 16 , 128 , 256 , 1024 , 2048 , 32768 so there are various power-of-2 jumps, but I haven't been able to figure out the overall pattern so that I can nail down the expression.Does anyone know of such an expression, or know of a good place to look to try to figure this out? If not, does anyone know if this is a fool's errand?Thanks for any help.

Question

Expression for binomial coefficient denominatorI&#039;m trying to find an analytical expression for the denominator of       (                            −          1                      /                    2                                      k                      )   in terms of k when the fraction is fully reduced.E.g., the first several such denominators, starting with   k  =  0, are   1  ,  2  ,  8  ,  16  ,  128  ,  256  ,  1024  ,  2048  ,  32768 so there are various power-of-2 jumps, but I haven&#039;t been able to figure out the overall pattern so that I can nail down the expression.Does anyone know of such an expression, or know of a good place to look to try to figure this out? If not, does anyone know if this is a fool&#039;s errand?Thanks for any help.

wayembesail · Accepted Answer

This is integer sequence A046161 in the On-Line Encyclopedia of Integer Sequences.You can find several formulas and details there.

vamosacaminarzi · Accepted Answer

We haveA result of Kummer states that the highest power of 2 dividing                      (                            2        k            k                      )             is the number of carries when adding n to itself in base 2, which in this case is easily seen to be the number of 1s in the binary representation of k. Hence the answer is        2          2      k      −              |            k                        |                2            , where       |    k            |        2   is the number of 1s in the binary representation of k.

Expression for binomial coefficient denominator I'm trying to find an analytical expression for the

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