What is the chemical formula for the synthesis of Hydroxyapatite (Ca10(PO4)6(OH)2)?
First, let me state the standard formula of hydroxyapaptite:

The term &quot;apatite&quot; applies to a group of compounds  with a general
formula in the form $\ce{M10(XO4)6Z2}$, where $\ce{M^2+}$ is a metal
and species $\ce{XO4^3-}$ and $\ce{Z-}$ are anions. The particular
name of each apatite depends on the elements or radicals M, X and Z.
In these terms, hydroxyapatite (HAp) has the molecular structure of
apatite, where M is calcium ($\ce{Ca^2+}$), X is phosphorus
($\ce{P^5+}$) and Z is the hydroxyl radical ($\ce{OH-}$). This is
known as stoichiometric hydroxyapatite and its atomic ratio Ca/P is
${1.67}$. Its chemical formula is $\ce{Ca10(PO4)6(OH)2}$, with $\ce{39 {%}}$ by
weight of $\ce{Ca}$, $\ce{18.5 {%} P}$ and $\ce{3.38 {%}}$ of $\ce{OH}$.

The anion radical can be replaced with fluorine/chlorine to become fluoroapatite/chloroapatite. This formula can also be written as $\ce{3[Ca3(PO4)2]· Ca(OH)2}$ which indicates three moles of calcium phosphate and one mole of calcium hydroxide which is basically the same as the Sigma-Aldrich's formula. The confusion arises since they have omitted the brackets, you didn't considered calcium phosphate as single entity for which you had a wrong formula. Brackets are generally not given but you have consider them as a single entity or you can even use brackets conveniently for your purpose. But do note that this formula doesn't prove that it is a basic salt but is actually a product of polymerisation:

According to Bassett, the solid existing in stable equilibrium at $\pu{25 °C}$., with solutions of a range from faintly acid to nearly pure
lime-water, is oxy- or hydroxyapatite, but, although its composition
may be indicated by the formula $\ce{3Ca3(PO4)2.Ca(OH)2}$, it is not to be
regarded as a basic salt, but rather as the salt of a polymeride of
phosphoric acid, $\ce{H11P3O13}$ (or $\ce{3H3PO4.H2O}$), with one hydrogen atom
un-neutralised (like orthophosphoric acid or pyrophosphoric acid). The slowness of
formation of hydroxyapatite favours the view that it is produced by
polymerisation.

Reference

Hydroxyapatite-Based Materials: Synthesis and Characterization
By Eric M. Rivera-Muñoz, 2011, DOI: 10.5772/19123
http://calcium.atomistry.com/basic_calcium_phosphates.html