recursion formula {#de}
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In this section, we derive some recursions that further leads to the results in Theorem \[th1\].

First recall the definition of $f_m(k)$, which is the number of $M$-pillars whose characteristic sequence satisfies the following conditions:

1.  The total length of the characteristic sequence is $4k$;

2.  The sequence starts with a zero and ends with a zero;

3.  The number “1” appears only in the even position in the sequence and the number “-1” only in the odd position;

4.  The total number of “1” in the characteristic sequence is $m$;

5.  The sequences are made with the following subsequence structure: $$\begin{aligned}
    & 1\underbrace{00}_{\text{two}}-1\nonumber\\
    &1\underbrace{000000}_{\text{six}}-1\nonumber\\
    &1\underbrace{0000000000}_{\text{ten}}-1\nonumber\\
    &\quad\quad\cdots\nonumber\\
    &1\underbrace{0000\cdots 00000}_{4k-6}-1~.\end{aligned}$$

Also, we define another $M$-pillar, whose characteristic sequence also satisfies the above condition $(1)(3)(4)(5)$, but with $(2)$ replaced by the following $(2)^{*}$:

$(2)^{*}$ The sequence starts with a zero and ends with three zeros.

We denote $g_m(k)$ as the number of such $M$-pillar satisfying $(1)(2)^{*}(3)(4)(5)$.

Master recursions related to $f_m(k)$ and $g_m(k)$
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In this subsection, we derive two master recursions related to $f_m(k)$ and $g_m(k)$.

Once a characteristic sequence with length $2k$ is given, we denote $S_n$ ($1\leq n\leq2k$) as the partial sums of its first 