A childhood dream

When I was a kid, I learnt (a+b) the whole squared and (a+b) the whole cubed formula. I would wonder why there is no formula for a^n+b^n.

Today, I am partially fulfilling this dream of mine with consecutive numbers. For, I have made a somewhat equivalent formula.

The formula is

(Kindly check the formula thoroughly before use. I have checked for two cases and found it okay.)

I am in love with this formula of mine as it mixes up consecutive numbers, factorials and binomials, all three. 

I am sure you will like the above expression.

Have a very nice day !!