An algebra problem by Ραμών Αδάλια

Algebra Level 4

Let ${ a }_{ n }=\sum _{ k=1 }^{ n }{ (k+2)\sum _{ j=1 }^{ k }{ \sum _{ i=1 }^{ j }{ i!({ i }^{ 2 }+i+1) } } }$ and ${ b }_{ n }={ a }_{ n+1 }-{ a }_{ n } .$ Compute $$b_4$$.


Notation: $$!$$ is the factorial notation. For example, $$8! = 1\times2\times3\times\cdots\times8$$.

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