Riemann-Roch and sum of powers

Let $X$ be a smooth projective $n$-fold and $L$ be a line bundle on $X$. We have Riemann-Roch formula,

when $n=1$,$chi(L)=chi(mathcal{O}_X)+mathrm{deg}(L)$,

when $n=2$,$chi(L)=chi(mathcal{O}_X)+frac{L(L-K)}{2}$,

when $n=3$, $chi(L)=chi(mathcal{O}_X)+frac{L(L-K)(2L-K)}{12}+frac{L.c_2(X)}{12}$,

the fomula resembles sum of powers

$sum_1^n k^0=n$, $sum_1^n k^1=frac{n(n+1)}{2}$ and $sum_1^nk^2=frac{n(n+1)(2n+1)}{6}$

Is this a coincidence or they are related in general?