# Problem 568: Reciprocal games II
Tom has built a random generator that is connected to a row of \$n\$
light bulbs. Whenever the random generator is activated each of the
\$n\$ lights is turned on with the probability of \$\\frac 1 2\$,
independently of its former state or the state of the other light bulbs.
While discussing with his friend Jerry how to use his generator, they
invent two different games, they call the reciprocal games: Both games
consist of \$n\$ turns. Each turn is started by choosing a number \$k\$
randomly between (and including) \$1\$ and \$n\$, with equal probability
of \$\\frac 1 n\$ for each number, while the possible win for that turn
is the reciprocal of \$k\$, that is \$\\frac 1 k\$. In game A, Tom
activates his random generator once in each turn. If the number of
lights turned on is the same as the previously chosen number \$k\$,
Jerry wins and gets \$\\frac 1 k\$, otherwise he will receive nothing
for that turn. Jerry's expected win after playing the total game A
consisting of \$n\$ turns is called \$J\_A(n)\$. For example
\$J\_A(6)=0.39505208\$, rounded to 8 decimal places. For each turn in
game B, after \$k\$ has been randomly selected, Tom keeps reactivating
his random generator until exactly \$k\$ lights are turned on. After
that Jerry takes over and reactivates the random generator until he,
too, has generated a pattern with exactly \$k\$ lights turned on. If
this pattern is identical to Tom's last pattern, Jerry wins and gets
\$\\frac 1 k\$, otherwise he will receive nothing. Jerry's expected win
after the total game B consisting of \$n\$ turns is called \$J\_B(n)\$.
For example \$J\_B(6)=0.43333333\$, rounded to 8 decimal places. Let
\$D(n)=J\_B(n)−J\_A(n)\$. For example, \$D(6) = 0.03828125\$. Find the 7
most significant digits of \$D(123456789)\$ after removing all leading
zeros. (If, for example, we had asked for the 7 most significant digits
of \$D(6)\$, the answer would have been 3828125.)