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)$a(n)$ is the number of permutations p$p$ of {1,..,n}$\{1,\ldots,n\}$ such that the minimum number of block interchanges required to sort the permutation p$p$ to the identity permutation is maximized.
1, 1, 5, 8, 84, 180, 3044, 8064, 193248, 604800$1, 1, 5, 8, 84, 180, 3044, 8064, 193248, 604800,\ldots$ (oeis.org/A260695https://oeis.org/A260695)
Consider the following harmonic numbers:
1 + 1/2 = ((1 + 2)(1))/2!$$1 + 1/2 = (1 + 2)\cdot 1/2!$$ 1 + 1/2 + 1/3 + 1/4 = ((1 + 2 + 3 + 4)(5))/4!$$1 + 1/2 + 1/3 + 1/4 = (1 + 2 + 3 + 4)\cdot 5/4!$$ 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 = ((1 + 2 + 3 + 4 + 5 + 6)(84))/6!$$1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 = (1 + 2 + 3 + 4 + 5 + 6)\cdot 84/6!$$ 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!$$1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8)\cdot 3044/8!$$
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)!$$1 + 1/2 + 1/3 + 1/4 + \cdots + 1/2n = (2n^2 + n)\cdot a(2n - 1)/(2n)!$$
If this hasn't been proven, then I will leave it as a conjecture.
Cheers,
Robert
