# Generators as Symbolic Exploration

Sep 18, 2025

number algorithms

python ruby figurate numbers live coding integer sequences gamma function

First Domain Extension via Gamma Function

During the development of $k$ -dimensional figurate number generators in my libraries figuratenum and figurate_numbers, I encoded several sequences using the factorial and its companions.

Both the rising factorial $n^{\overline{k}}$ and the classical factorial, interpreted as a special case of the falling factorial $k^{\underline{k}}$ , are perhaps at their core a very natural way of counting dimensions and constructing geometric universes through monotonic multiplications along complementary paths.

Yet, seeking a different perspective to address my concerns, I often think of sequences as an enriched, serpentine wave connecting the discrete world of counting with the continuity of analysis (inspired by Artin’s book). The order in which I present the following winding identities simply reflects my personal taste.

H_k(n)=\frac{n(n+1) \cdots (n + k - 1)}{k(k-1)(k-2) \cdots 1}

= \frac{ \prod_{i=0}^{k-1} (n + i)}{ \prod_{i=0}^{k-1} (k-i)} = \frac{n^{\overline{k}}}{k^{\underline{k}}}

= \frac{\dfrac{\Gamma(n + k)}{\Gamma(n)}}{\Gamma(k + 1)} = \frac{\Gamma(n + k)}{\Gamma(n) \, \Gamma(k + 1)}

= \frac{n^{\overline{k}}}{k!} = \binom{n + k - 1}{k}

= \frac{ \int_0^\infty \tau^{n+k-1} e^{-\tau} \ d\tau} {\left( \int_0^\infty \tau^{n-1} e^{-\tau} \ d\tau \right) \left( \int_0^\infty \tau^k e^{-\tau} \ d\tau \right)}.

All these expressions unite in the object called the k_hypertetrahedron $H_k(n)$ , following Figurate Numbers (2012), where combinatorics, geometry, and analysis converge.

On the purely combinatorial side, $H_k(n)$ is tightly connected to integer partitions. In a geometry view, it counts the integer points $(x_1, \dots, x_k) \in \mathbb{N}^k$ satisfying the equation $\sum_{i=1}^k x_i = n$ , forming a discrete $k$ -simplex. This often reminds me of a modified Tonnetz, where simplices tile symbolic space.

Finally, seen through the lens of the Gamma function it becomes malleable and extends to fractional or complex values $H_k(z)$ , provided the constraints on the real part are respected.

Towards a Symbolic Extension in Each Term

I have thought of building generators as a form of navigation. It is not just about producing sequences, but about following a winding path through ideas, like a symbolic river.

Although implementations may appear rigid for the sake of efficiency, as I mentioned in the previous post, I also included book-style versions. That choice made me reflect on the possibility of creating broader, more elegant functions.

def k_dimensional_hypertetrahedron_from_book(k: int) -> Generator[int]:
    """Book definition. For reference only."""
    delta = 1
    while True:
        yield rising_factorial(delta, k) // factorial(k)
        # One alternative under the analytic mantle:
        # yield pochhammer_symbol(delta, k) / gamma_func(k + 1)
        delta += 1

I always keep these ideas in mind as the conceptual foundation for designing generators more deeply rooted in different mathematical spaces. In that spirit, I hold on to this question:

If the symbolic expressions that define the figurate number are themselves the first elements generated by that figurate number: what new definition, equivalent yet amplified, whether combinatorial, geometric, algebraic, or analytic, emerges in a given dimension $k$ and at the $n$ -th term?