In simple terms what is the sequence 220,171700,167167000,166716670000 a part of?

Question

I know that

<1,2,3,...,10>$\cdot$<1 0,9,8,...1>=220

<1,2,3,...,100>$\cdot$<100,99,98,...,1>=171700

<1,2,3,...,1000>$\cdot$<1000,999,998,...,1>=167167000

<1,2,3,...,10000>$\cdot$<10000,9999,9998,...,1>=166716670000 `

And 2,17,167,1667 is a part of the OEIS A126109 (5*10^n+1)/3. But in simple terms what is the sequence 220,171700,167167000,166716670000 a part of?

Answer 1

The $n$th term of your sequence is the $10^{n}$th term of A000292, that is your $n$th term is $$\frac{10^n(10^n+1)(10^n+2)}{6} $$

Answer 2

We have $$<1,2,3,...,n> \cdot <n,n-1,n-2,...,1> $$ $$= \sum_{i=1}^n (n+1-i)i = \sum_{i=1}^n i(n+1) - \sum_{i=1}^n i^2$$ $$= (n+1) \frac{1}{2}n(n+1) - \frac{1}{6}n(n+1)(2n+1) = \frac{1}{6}n(n+1)(n+2).$$

Replacing $n$ by $10^n$ yields the general term of your sequence.