Is there a symbol for a factorial with a modular congruence condition?

Question

I'm solving ODE's using power series and I'm often getting coefficientes that involves terms like \begin{equation} \prod_{n=1,\:\:n\not\equiv1\pmod{3}}^{3k}n=2\cdot 3\cdot 5\cdot 6\cdot 8\cdot 9\cdot \dots\cdot (3k-1)\cdot 3k. \end{equation}

Is there any symbol for such a product? I've heard of double factorials. Would a "triple factorial" be standard notation? Any suggestions on how to write this? The product formula above is too big for my purposes.

Answer 1

Rewrite it as the ratio of two products. The numerator will obviously be none other than $(3k)!$

while the denominator will be

$$\begin{align}\prod_{j=0}^{k-1}(3j+1) \quad&=\quad 3^k~\prod_{j=0}^{k-1}\bigg(j+\dfrac13\bigg) \quad=\quad \dfrac{3^k}{\Gamma\bigg(\dfrac13\bigg)}\cdot\Gamma\bigg(\dfrac13\bigg)\cdot\prod_{j=0}^{k-1}\bigg(\dfrac13+j\bigg) \quad=\\&=\quad \dfrac{3^k}{\Gamma\bigg(\dfrac13\bigg)}\cdot\Gamma\bigg(k+\dfrac13\bigg). \end{align}$$