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So, Sejuani was the first champion I picked up in the jungle to learn it at first. Her kit is very strong with an ultimate which is game changing and can win a whole fight on its own. Its practically an Amumu ultimate, but ranged. Sejuani is also mainly known for her incredible crowd control and tankiness, but is also quite versatile being capable of going full ap and literally one shotting enemies with a combo. Instead of a usual boring tank build, I will talk about a surprisingly insane full ap build.

Pros

-Brings a lot of utility
-Has insane damage mid and late game
-Quite safe early game with passive bringing tankiness.
-People don't expect her damage.
-Can turn fights with her ultimate.
-Some of the best cc in the game.
-Can go tanky if the game is going badly.
-Doesn't need many items for damage.
-Insanely fun.

Cons

-Squishy and easily kited.
-Poor early game clear.
-Scared of invades.
-Extremely squishy when AP.

Flash is a great spell on Sejuani. It allows her to be safe early game and engage from really for with combination of her Q. Almost every champion takes this and there's a reason why.
Smite because you're jungling....
Ghost is probably the only other spell which is viable other than smite. But I dislike the Idea of ghost on full ap Sejuani. I however do think it is viable on tank Sejuani.
Don't see it happening, especially since Sejuani is not a good duelist for split pushing and her other spells are just more valuable.
I could see this working. It'd be great for ganks, but sejuanii excels in team fighting, so pinning down one target is not your priority especially if you're playing this build.
No. You don't need the early kill pressure or snowball since you are best in the mid-late game and will scale even just farming.

You are not an ADC, you get damage from your ability combo and not from auto attacks. This applies both early game ganking and teamfights.
You do AoE damage. While Feast may help with sustain, Expose Weakness allows your team to kill everyone very quickly which will win you more games. Feast may be taken if you are scared of invades, but I prefer Expose Weakness
You want damage. This adds to your burst, while spell vamp has potential, you will mostly be engaging which is when you're at full health. You won't heal hp at full health.
Helps with your clears. Wanderer can also be good for engaging, but I think you'll want this mastery more for your clears.
The extra time with your buffs helps quite a lot. Since your he jungler, you have all the right to have both your buffs at any point of the game.
You don't really go mana hungry especially with your blue buff. Late game mana is no problem whatsoever, so more damage is better.
You won't even be in a lane, so Bandit is pretty useless.
You're Ap. It's simple, you don't do ad, so you want magic penetration.
This adds to your burst and helps with killing in early ganks. The other option is Stormraider's Surge and to be honest, I kinda like the idea of that movement speed after bursting everyone from 100 to 0, but Thunderlord's Decree will work through the whole game and you already have tons of cc, so movement speed will probably not be needed. However, you may want to take this against teams with kite.

Good Stats, plus a great active. This item is amazing for adding to your damage and just for the utility it brings. The magic bolts fired through the active are also great because of how easy it is to land after using your ultimate. Overall good utility and burst help early. The cheap price is amazing too.
Another item which gives great stats, but the passive is just perfect. You can laugh and watch as your enemies burn while being frozen in place. They passive is so great since your w and e slow everyone and your ultimate is also aoe. Still item is still good when you go tank.
Can't live without these just like any champion.How you upgrade them are up to you, but your default choice should be Sorcerer's Shoes or maybe Ionian Boots of Lucidity. Mercury threads is mainly for tank, but if they just cc you crazy, this is an option. ninja tabi's are for tank Sejuani against those teams (EG: Jax top, Master Yi jungle, Yasuo mid, Jinx adc).

Other Options

They have tons of ap poke from champions like Lux and Ziggs and they have champions building magic resist. This is not your go to magic penetration item, but it should help with it.
This is usually if you're doing well. Build this against teams which cannot burst you down after you engage. This brings serious amounts of damage and is really fun to have.
You already know when to build this.
When they have champions which are just so annoying and heal a lot. This is actually quite common in the meta now. (Eg: Swain, Hecarim, [Vladimir]].)
When you have a lot of ap, usually as a 4/5/6th item. Often when you have extra gold, you can sell an item for this just for the damage.
They are building to counter you with tons of magic resist. This will help you continue to burst them down even after they have something like a Spirit Visage.
This is also quite a fun item. Build it against champions like Zed or just assassins. This is also quite a great buy if your team has good follow up and damage since you can engage, use Zhonya's Hourglass and let your team finish them off.
I personally dislike this on full ap Sejuani, since it takes away from your burst and since you go in melee, it won't mean much after if the enemy team is not all dead, since they can just finish you off when you re spawn since you're a squishy.

Early Game

Stay safe is probably my number one rule. You just need to get 2 or 3 items which is when you will be able to demolish everyone. You don't have the fastest clear, so try to keep up to the opponent and get the free minions/exp when your allies leave lane. Though you aren't an early champion, your ganks are still some of the deadliest. If you can chain cc someone, they're almost guaranteed dead. How you should gank is usually initiate with Arctic Assault and auto, then flail of the north winds. By now hopefully your laner has used some of their own cc leaving them stuck in place. Save your Permafrost for when they start to run since it will help chase.

Mid Game

Now is your time to shine. Show everyone a magic trick by making the enemies disappear. They won't expect that ultimate to do so much damage with that Liandry's Torment. You should tell your team to group and teamfight which is Sejuani's strongest ability. Wait for the perfect moment and hide out of vision like a Fiddlesticks with your Oracle Lens and when they are all grouped up, throw your Glacial Prison and Arctic Assault into the fight. Follow that up with using your hextech protobelt and Permafrost. You don't need the slow, just the burst damage for your Permafrost. Activate your flail of the north winds and watch them get deleted.

Late Game

Now, if the game hasn't ended you have 2 choices. Either convert to tank which is the safer play. Or try to make picks/win a teamfight to win. Both are very realistic options for Sejuani. If you chose to stay full ap, do what you were doing in the mid game, but safer. Don't simply engage, make sure you have both vision and backup for when you engage. Once you go in as Sejuani, you're likely not coming out, so a Zhonya's Hourglass may be the item you're looking for to live after the fight. Focus down any important targets and hopefully you'll win.