Homology of compact non-orientable surfaces

Statement
Suppose $$k$$ is a positive integer. We denote by $$P_n$$ (not standard notation, should try to find something) the connected sum of the real projective plane with itself $$n$$ times, i.e., the connected sum of $$n$$ copies of the real projective plane.

Unreduced version over the integers
We have:

$$H_k(P_n;\mathbb{Z}) = \lbrace\begin{array}{rl} \mathbb{Z}, & k = 0 \\ \mathbb{Z}^{n-1} \oplus \mathbb{Z}/2\mathbb{Z}, & k = 1\\ 0, & k \ge 2 \\\end{array}$$

Reduced version over the integers
We have:

$$\tilde{H}_k(P_n;\mathbb{Z}) = \lbrace\begin{array}{rl} 0, & k = 0 \\ \mathbb{Z}^{n-1} \oplus \mathbb{Z}/2\mathbb{Z}, & k = 1\\ 0, & k \ge 2 \\\end{array}$$

Unreduced version over a module
If we consider the homology with coefficients in a module $$M$$ over a ring $$R$$ where 2 is invertible, then we have:

$$H_k(P_n;M) = \lbrace\begin{array}{rl} M, & k = 0 \\ M^{n-1}, & k = 1\\0, & k \ge 2 \\\end{array}$$

Homology groups with integer coefficients in tabular form
We illustrate how the homology groups work for small values of $$n$$. Note that $$H_2$$ is zero and all higher $$H_p$$ are zero.

Related invariants
These are all invariants that can be computed in terms of the homology groups.