1. A strong centralized state that can fund such an thing. (I.E. Eastern Roman Empire)
Depending on your definition of centralized, it may be not the case. A bureaucratic state should be enough, having several feudal or pseudo-feudal identities within his borders but able to raise and manage taxes on the realm scale.
2. Inspiration from the Roman Empire to create a Professional Army (I.E. Matthais Corvinus)
Not really. IOTL western standing armies were formed from former mercenary companies. In fact, for the contemporary eye, they were such. Only that the king maintained their presence (while in clearly reduced numbers) with himself at their head (delegating his power to courtier) instead of an independent capitain.
French
compagnies d'ordonnance's soldiers, for instance, while clearly a standing army, had its soldiers buying their own equipment, inheritence from old
routiers.
In fact, while standing armies re-appeared in Europe only at the middle of XVth century,
professional armies appeared much earlier with mercenary armies, whom soldiers lived from war.
(That said, I'm pretty sure that Corvinus' Black Army had the same roots).
3. Possible earlier discovery of gunpowder that would eventually in the decline of the Feudalistic Knights.
Gunpowder was used in the XIIIth century, and didn't caused its decline.
Earlier cannons (up to the XVth century) were quite inneficient, at the point they were used as psychological weapons rather than siege (mechanical artillery being more efficient).
Two factors allowed cannons to be a decisive changes.
1) Metal projectiles : before that, balls were made of stone, with a bad penetration power, not making late medieval fortification that threated. When mettalurgic knowledge progressed, and made this change happen, the situation was different.
But even there, castles were modified : larger walls, cannons on them,
new fortifications. The thing is, it costed quite a lot, and only great lords or important cities could fund them, and only great lords could do that.
That said, the tendency to great lords to absorb their weakest neighbours was already quite established by the XIIIth century. It only quikened events there.
2) Establishment of standing armies.
There I think you got the cause/effect wrong. As said, gunpowder artillery existed before the XVth century but never was a real threat.
But at the moment where standing armies with strong artillery components (that only great lords could fund) existed instead of relativly isolated and limited previously used, it became hard to resist an actual artillery barrage, definitely chaging the look of war.
So, gunpowder is less a premise for standing army, than a cause that admittedly influenced back its premise.
And (using again French exemple) even when the biggest lords were crushed or tamed, you still had an important feudal component in war, alongside standing armies (that went quite neglected for a time, especially during early Italian Wars)
The same ordinances that created standing compagnies also re-organized feudal levies and systematized them.
Basically, feudal levies and standing armies weren't opposite, and french kings (whom power still dependend from feudal institutions, depsite the strengthening of their authority) couldn't erase them.