I will outlay a few observations applying to the harmonic numbers that may be interesting to prove (if it hasn’t already been proven).

From the Online Encyclopedia of Positive Integers we have:

a(n) is the number of permutations p of {1,..,n} such that the minimum number of block interchanges required to sort the permutation p to the identity permutation is maximized.

1, 1, 5, 8, 84, 180, 3044, 8064, 193248, 604800 (oeis.org/A260695)

Consider the following harmonic numbers:

1 + 1/2 = ((1 + 2)(1))/2!

1 + 1/2 + 1/3 + 1/4 = ((1 + 2 + 3 + 4)(5))/4!

1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 = ((1 + 2 + 3 + 4 + 5 + 6)(84))/6!

1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 = ((1 + 2 + 3 + 4 + 5 + 6 + 7 + 8)(3044))/8!

and so on.

Therefore, the following generalization suggests itself:

1 + 1/2 + 1/3 + 1/4 + … + 1/2n = ((2n^2 + n)a(2n – 1))/(2n)!

If this hasn’t been proven, then I will leave it as a conjecture.

Cheers,

Robert