I am FoxKorn. I am actually only a Silver rank player, but i am pretty good at math and came up with this build. A challanger player named Grape came up with nearly the same build on his own and he took AP Sejuani to Master. So he shows that it works.
I put a lot of work into this aswell as some Excel calculations. I hope you can learn a bit about Sejuani and how League works using this build. And of course have a lot of fun. (I am not native English so excuse me for some typos you might find^^" )

Short answer: Because it works.

I love playing Off-Meta Champions and joined the Deep League community. After joining i wanted to make my own Off-Meta mathematically correct build and came up with this. Join the Deep League Discord, if you want to explore more mathematically builds.

Main

	Electrocute is a must since Sejuani can proc it always with a simple Q -> W Combo
	It's good for poking the enemy and synergizes with Sejuani's W
	Since you want to oneshot people you are going for Kills/Takedowns which makes stacking Eyeball Collection quite easy.
	Sejuanin's ultimate is the best tool she has, so we want to use it as much as possible.

Secondary

	It's just like Cheap Shot. Good for poking the enemy and synergizes with Sejuani's W
	Since Sejuani has quite some cooldowns it's good to take a bit ability haste with you. Also with lvl 11 takedown reduce remaining cooldows of basic abilities by 20%. Which is nice since this is a oneshot build.
	Take Nullifying Orb instead of Transcendence if you need a bit safety against some heavy magic dmg. But monstly it's not needed to go defensive with Runes.

Alternative / Defensive

	It's not realy ideal to go defenvise here, but depending on your playstyle this could work out better for you in some matchups. The HP from Overgrowth is good defensive and offensive since Sejuanis W also scales with HP.
	Bone Plating is like a little boost for Sejuanis Passive and keeps you safe from early burst.

Here is not much to say. We want to oneshot people so we go Ignite for dmg and Flash, because who doesn't takes Flash these days.

Introduction

Sejuani has a lot of base-damage on her spells because she is mainly a tank. This is what this build kind of exploits. The Ability Power scaling isn't as good as on the Meta-Mages, but it's there and Sejuani can work with it too. I spend quite some time in practice tool and Excel spreadsheets just to find out that focusing on Magic Penetration is better than focusing on Ability Power. Keep that in mind for the items when you play AP Sejuani. On top of that Sejuani can make use of the HP scaling on the W Winter's Wrath with some HP on the Items.

Starting Items

Doran's Ring and Health Potion is a good start which isn't as "snowbally", but good for the first couple rounds (until you learn how to play Ap Sejuani) and against hard matchups.

Dark Seal and Refillable Potion are more "snowbally", but also makes you a target. Every enemy jungle should notice your items and try to punish you for that. Which makes your laneing harder, but since Sejuani has her passive Fury of the North and her Q Arctic Assault to escape, she is actually pretty save. (Since you play an Off-Meta Champion the Jungle will most likely try to get you anyways, so building these two items actually doesn't change much.) So if you are a bit experienced with this build you should be able to run these two items almost every game.

First base

	It gives you HP and AP, but the passive is the important part here. Early this item is quite strong for sudden burst and goes well with the high base-damage from Sejuani. Since Hextech Rocketbelt, Horizon Focus and Shadowflame use Hextech Alternator as a component you should build it everytime you can.
	If you started with Doran's Ring it's not to late to go Dark Seal if you are confident enough now.
	Also don't forget to buy some lvl 1 Boots, if you can.

Mythic

I recommend Hextech Rocketbelt, because it has the Magic Penetration. Since Sejuani's Base-ability-damage is so big and the AP scaling is kind of small, it's actually better to build full Magic Penetration instead of mostly Ability Power. It also adds a little dash which can be usefull since the Q Arctic Assault has quite some cooldown.

Some people apparently play Everfrost on Sejuani. While with Hextech Rocketbelt Sejauni deals more damage Everfrost has the root, which can be used to hold the enemy in place to hit some Autoattacks, which grant you some Permafrost stacks. It's a good item, if the enemy has his own escape tools and you don't want them to move at all.

Some math: At full-build you will deal more dmg to enemys with Everfrost if they are below 65 Magic resistance, but more dmg with Hextech Rocketbelt if they have more than 65 Magic resistance. This border of 65 is actually lower before Rabadon's Deathcap and since you won't build 6 Items in most of your games, you deal less damage with Everfrost most of the time.

Second and Third Items

Sorcerer's Shoes because of the magic penetration.

Horizon Focus is the core item of Ap Sejuani. The 10% bonus damage activate on every skill since all skills of Sejuani have immobilizing effects. So this item boosts nearly all damage of Sejuani by 10%, which means the strong base-damage of all skills is scaled up. (Note: The W Winter's Wrath has a knockback with the first cast, which is an immobilizing effect, but the second part is a slow, which doesn't count as an immobilizing effect. Slows are only impaired movement effects.)

Fourth and Fifth Items

	Void Staff because of the magic penetration. If the enemys didn't bought any magic resistance you can delay Void Staff and buy it as your fifth item, but mostly you go with it as your fourth item.
	My cauculations and the Practice Tool say that you deal the most dmg with Shadowflame.
	If you bought Dark Seal then you should probably build your Mejai's Soulstealer if things are going well. Obviously it's a stong item at max stacks and since this build can oneshot any non-tank enemy you will get there fast. But the enemy will probably try everything to stop you at this point if they aren't trying this anyways since you oneshot their carries. (Note: Mejai's Soulstealer replaces Rabadon's Deathcap so you should buy Void Staff and Shadowflame anyways.)
	Some people play Lich Bane on Sejuani and seem to make it work. Since Sejuani always autoattacks after the Q Arctic Assault you can proc the Lich Bane once but that's it. You won't wait for 1,5 seconds after the autoattack just to use W Winter's Wrath to proc Lich Bane again and when you are pokeing with W Winter's Wrath you won't be able to land an autoattack, since you will mostly use it on a long range. It's better to go with Shadowflame because your overall damage is higher if you don't proc Lich Bane three times.
	Since the changes on Cosmic Drive you need to hit the enemy with 3 seperate attacks, which means those attacks deal not extra damage and thats probably the most of your damage. Also it doesn't give the big amount of Ability Haste as it used to give you. It became a bad item to build on Sejuani.

Last Item

Your last item is Rabadon's Deathcap if you don't have Mejai's Soulstealer, because it's the item that gives you at this point the biggest dmg boots. If you lose your Mejai's Soulstealer stacks and you have enough money to get Rabadon's Deathcap it's a good decision to switch. Usually the game should be already over before you would be able to build this.

	Arctic Assault will be maxed second because it's your all-in / cc tool, use Arctic Assault in lane only if you are ready to all-in and don't waste it since it's also a great way to disengage/escape and avoid ganks from the enemy jungler. You can also cast Winter's Wrath while dashing for a good and easy all-in. ( Arctic Assault does not apply Permafrost stack, but Sejuani Autoattacks after the Arctic Assault.)
	Winter's Wrath is maxed first because it's the only ranged skill Sejuani has and deals a good amount of physical damage. It also applies with both animations a Permafrost stack . It's the skill you will be using most for some small trades and pokes.
	Permafrost is only available after you get 4 stacks of Permafrost on the enemy. Permafrost stuns the enemy and interrupts dashes (it's a knockdown) other melee champions in your near can apply Permafrost stacks with Basic attacks too. So ganking Sejuani is amazing and roaming with Sejuani is just as good as getting ganks. ( Permafrost can't be applied on an enemy Champion for 8 seconds, if the enemy was stunned by Sejuani => after hitting your ultimate Glacial Prison you won't be able to use your on this target. Just to be clear, Arctic Assault is a knockup and doesn't count as a Stun here)
	Glacial Prison deals a lot of Dmg after it traveled 400 units and stuns the enemy hit. Also slows everyone in the area around the hit enemy and deals to them the same dmg, if they stay in the slow-field. So if the enemy botlane doesn't look on their maps, your team will get some easy kills with that.
	Don't forget your passive. The Passive is a good little safety tool which gives you resistances and slow immunity for 3 seconds if you get dmg ( resets after 12 sec of getting no dmg at all).

Level 1-5

Early you want to poke the enemy with W Winter's Wrath and push the wave a bit to get lvl 2 first. If you hit the enemy already about two times, you can go all-in at lvl 2 with a Q Arctic Assault->W Winter's Wrath-> Ignite-Combo (or better but a bit harder W->Q->AA + Ignite) if the enemy doesn't have lvl 2. This works mostly because the enemy doesn't expect such a burst out of nowhere which is why this build has Electrocute and Cheap Shot as runes. Some enemys will respect your lvl 2 and play safe. You shouldn't force a kill if you can't or you die yourself too.

That's how the laning should continue. Poke with W Winter's Wrath whenever you can. Only all-in with Q Arctic Assault if you can get the kill. Q Arctic Assault should be always ready to escape, since the enemy jungler will probably try to get you. Also use the passive Fury of the North for trades to go out ahead anytime you can.

Falling a bit behind isn't such a big problem, since Sejuani has a lot of base-dmg and deals enough dmg mostly because of her level and base-damage. Missing some minions isn't as bad as on other champions.

Level 6

With your ultimate Glacial Prison you are ready to oneshot the enemy. When you land an ultimate with 400 untis distance you should be able to kill any squishy champion. It doesn't matter if it's your midlane opponent or the botlane/toplane.

You should always roam with your ultimate Glacial Prison up, which is an amazingly strong ganking tool.
If you chose to roam than always push your lane out with W Winter's Wrath so you won't miss any minions.
Don't be afraid of ganking toplane. You might not be able to oneshot a bruiser/tank, but with your jungler/toplaner you should have enough dmg and Sejuani brings a lot of CC with her kit.

Roaming

You should look for roaming opportunities everytime you can. Even without your ultimate Glacial Prison Sejuani has enough CC to get some kills/assists and don't forget that other nearby friendly melee champions grant E Permafrost stacks. This should benefit you and your whole Team.

Since Sejuani is a jungler she is good a ganking and that's what you should do, thats why this build is good midlane. Abuse this as much as possible.

If done correctly your team should be ahead now and getting every objective.

Remember to not jump into to many enemys since you mostly just oneshot one person and you aren't very tanky.
Also clearing some wards while roaming is strongly recommended. So don't forget to buy a Control Ward.

Don't jump in first

You might look big and tanky, but you aren't.
Never go in first, because you will most likely die.
Remember that your ultimate Glacial Prison is like a little Nunu & Willump ultimate. Once you throw it far enough and hit an enemy or even not it will slow and exlpode after 2 seconds. You can use this to let your team engage and follow after your tank/bruiser tanked most of the enemys CC and burst.
It's ideal if you can flank the enemys from behind/the side and get a pick on an enemy carry while your team engages from the front.

Teamfights are probably to most tricky part of this Off-Meta build, but it's possible and if you played correctly your team should be ahead anyways because of your ganks so Teamfights shouldn't be a problem for your Team.

- After some calculations i am sure, with this build you should have enough damage to oneshot non-tanks even when they buy one or even two Magic resist-Items.
- Your ultimate Glacial Prison is almost a secure kill for you or your Teammates. That's why Dark Seal is something you should build regular on Seju and Arctic Assault gives you a lot of safety in lane as well as your passive Fury of the North.
- The Ultimate Glacial Prison is also and amazing tool for Teamfights. You should never engage first in a Teamfight, since you aren't tanky at all even tho Seju looks big and strong.
- Just try to kill your lane or roam to kill other lanes especially everytime your ultimate is up.
- Since Seju has some good base damage on those spells, it's not that so bad if you fall behind a bit.
- I recommend taking Flash and Ignite and going mid with this build.