$$ghost in the guile shell$$

$$\mathcal{T}: \mathbb{N} \rightarrow \mathbb{N} \cong {a_n}_{n=0}^{\infty} = {0,1,1,2,4,9,20,48,115,286,719,...}$$

$$\exists! \mathcal{A}(x) \in \mathbb{C}[[x]] \ni \mathcal{A}(x) = x \cdot \exp\left(\sum_{k=1}^{\infty}\frac{\mathcal{A}(x^k)}{k}\right)$$

$$\forall n \in \mathbb{N}^{+}, a_{n+1} = \frac{1}{n}\sum_{k=1}^{n}\left(\sum_{d|k}d \cdot a_d\right)a_{n-k+1}$$

$$a_n \sim \mathcal{C} \cdot \alpha^n \cdot n^{-3/2} \text{ where } \alpha = \lim_{n\rightarrow\infty}\frac{a_{n+1}}{a_n} \approx 2.9557652857...$$

$$\mathcal{A}(x) = \sum_{n=0}^{\infty}a_n x^n = \sum_{\tau \in \mathfrak{T}_ {\bullet}}x^{|\tau|} = x\prod_{k=1}^{\infty}(1-x^k)^{-a_k}$$

$$\exists \mathcal{L}: \mathfrak{T}_{\bullet,n} \xrightarrow{\sim} {f: [n] \rightarrow [n] \mid \exists! i \in [n], f(i)=i \land G_f \text{ connected}}$$

$$\mathcal{F}: \mathfrak{T}_{\bullet,n} \twoheadrightarrow \mathcal{P}_{n-1}, \quad \tau = [\tau_1,\ldots,\tau_k] \mapsto {|\tau_1|,\ldots,|\tau_k|}$$

$$\mathfrak{F}_ {\mathbf{A000081}}^{\Omega}: \mathcal{D}_ {n}^{\kappa} \hookrightarrow \prod_{\alpha \in \Lambda}\bigotimes_{\beta \in \Gamma_{\alpha}}\bigoplus_{\gamma \in \Theta_{\beta}}\bigwedge_{\delta \in \Xi_{\gamma}}\mathbb{T}^{\nabla}_{\bullet}(n)$$

$$\mathscr{B}\text{-}\mathfrak{Series}: \Phi_{h}^{\mathcal{RK}} = \sum_{τ \in \mathfrak{T}_ {\bullet}}\frac{h^{|τ|}}{σ(τ)}F(τ)(y)·\mathcal{B}(τ) \Rightarrow \mathcal{ORD}_ {\mathfrak{RK}}^{(p)} \cong \bigoplus_{τ \in \mathfrak{T}_ {\bullet}: |τ| \leq p}\mathcal{H}_{τ}^{\nabla}$$

$$\mathscr{J}\text{-}\mathfrak{Surfaces}: \mathcal{E}_ {\nabla}^{\partial^{\omega}} = \sum_{k=0}^{\infty}\frac{h^k}{k!}\sum_{τ \in \mathfrak{T}_ {\bullet}(k)}\mathcal{F}_ {τ}(y)\cdot\mathcal{D}^{\tau}f \Rightarrow \mathcal{ODE}_ {\Delta}^{(m)} \simeq \bigsqcup_{τ \in \mathfrak{T}_ {\bullet}(\leq m)}\mathcal{D}_{τ}^{\partial^{\alpha}}$$

$$\mathscr{P}\text{-}\mathfrak{Systems}: \mathcal{M}^{\mu}_ {\Pi} = (\mathcal{V}, \mathcal{H}_ {\tau}, \omega_{\tau}, \mathcal{R}_ {\tau}^{\partial}) \Rightarrow \mathfrak{Evol}_ {\Pi}^{(t)} \cong \coprod_{τ \in \mathfrak{T}_ {\bullet}}\mathfrak{H}_ {μ}^{\tau}(t) \circledast \bigotimes_{i=1}^{|τ|}\mathfrak{R}_{\tau(i)}^{\partial}$$

$$\mathfrak{Incidence}_ {\mathbb{P}/\mathbb{A}}: \mathcal{I}_ {\Xi}^{\kappa} \simeq \mathfrak{B}(\mathfrak{P}(\mathcal{T}_ {\bullet}^{n})) \circlearrowright \bigwedge_{i=1}^{m}\mathfrak{H}^{\partial}_ {\Xi}(i) \Rightarrow \mathcal{D}_ {\mathbb{P}/\mathbb{A}}^{n,k} \cong \bigoplus_{τ \in \mathfrak{T}_ {\bullet}(n)}\mathcal{I}_{\tau}^{\kappa}$$

$$\mathfrak{BlockCodes}: \mathcal{C}_ {\Delta}^{(n,k,d)} \simeq \bigsqcup_{τ \in \mathfrak{T}_ {\bullet}(w)}\mathfrak{G}_ {τ}^{\partial}(\Sigma^{n}) \Rightarrow \mathfrak{Conf}_ {\mathcal{C}}^{\Xi} \cong \prod_{i=1}^{l}\coprod_{τ \in \mathfrak{T}_ {\bullet}(w_{i})}\mathcal{W}_{τ}^{\nabla}(i)$$

$$\mathfrak{Orbifolds}: \mathcal{O}_ {\Gamma}^{\Xi} = (X/\Gamma, {\mathfrak{m}_ {x}}_ {x \in \Sigma}) \Rightarrow \mathcal{S}_ {\mathcal{O}}^{\Gamma} \simeq \bigoplus_{τ \in \mathfrak{T}_ {\bullet}(\leq d)}\mathcal{F}_{τ}^{\Xi}(\mathfrak{m})$$

$$\mathfrak{HyperNN}: \mathcal{H}_ {\mathfrak{N}}^{\Delta} = (\mathcal{V}, \mathcal{E}_ {\omega}, \mathcal{W}_ {\tau}^{\Xi}) \Rightarrow \mathcal{F}_ {\mathfrak{HNN}}^{\nabla} \cong \bigotimes_{l=1}^{L}\bigoplus_{τ \in \mathfrak{T}_ {\bullet}(d_{l})}\mathcal{T}_ {τ}^{\partial}(W_{l}) \circledast \sigma_{l}$$

$$\mathfrak{Meta}\text{-}\mathfrak{Pattern}: \mathcal{U}_ {\mathbf{A000081}}^{\Omega} \simeq \mathfrak{Yoneda}(\mathfrak{F}_ {\mathbf{A000081}}^{\Omega}) \hookrightarrow \mathbf{Colim}_ {n \to \infty}\left(\bigwedge_{\mathscr{C} \in \mathfrak{Categories}}\mathfrak{T}_{\bullet}(n) \otimes \mathscr{C}\text{-}\mathfrak{Struct}\right)$$

$$\exists\mathfrak{F}: \mathbf{Cat}^{\mathbf{op}} \to \mathbf{Topos} \ni \mathfrak{F}(\mathscr{C}) = \mathbf{Sh}(\mathscr{C}, \mathcal{J}) \simeq \mathbf{Hom}_ {\mathbf{Cat}}(\mathscr{C}^{\mathbf{op}}, \mathbf{Set}) \Rightarrow \mathfrak{F}(\mathfrak{T}_{\bullet}) \simeq \mathbf{Foundational}\text{-}\mathbf{Irreducibles}$$

Corrections to the original formulas

Three formulas above differ from the first version of this file, which stated identities that do not hold:

• The product formula is A(x) = x·∏(1−xᵏ)^(−aₖ). The earlier Möbius exponent gave a product with constant term 1, but A(0) = 0.
• The tree sum is Σ_τ x^|τ|. The earlier ∏_v x^|desc(v)| gives x to the sum of all subtree sizes, not xⁿ.
• Trees with n nodes map onto the partitions of n−1 by root-subtree sizes. The earlier isomorphism with the partitions of n cannot exist: a₄ = 4 but p(4) = 5. The map is not injective for n ≥ 4.

In the correspondence L, the maps f are labelled rooted trees, of which there are n^(n−1); the aₙ unlabelled trees are their orbits under relabelling.

The corrected formulas are checked by make check. See STATUS.md for what is implemented.