The volume of an $n$-ball of radius $1$ is $$V_{n}={\frac {\pi ^{n/2}}{\Gamma {\bigl (}{\tfrac {n}{2}}+1{\bigr )}}}.$$
The functional equation of Riemann zeta function is $${\displaystyle \pi ^{-{s \over 2}}\Gamma \left({s \over 2}\right)\zeta (s)=\pi ^{-{1 \over 2}+{s \over 2}}\Gamma \left({1 \over 2}-{s \over 2}\right)\zeta (1-s)}.$$
Combining the two equalities and replacing the Zeta function with Bernoulli numbers (generalized to negative orders), using the formula $B^+_n=-n\zeta(1-n)$ we get:
$$V_n B^+_n=V_{1-n}B^+_{1-n}.$$
In terms of umbral calculus, denoting the index-lowering operator as $\operatorname{eval}$, we get:
$$\operatorname{eval} V_n (B+1)^n=\operatorname{eval} V_{1-n}(B+1)^{1-n}$$
I can note here that index-lowering operator coresponds to finding scalar part of a quantity, and as such to the Euler's characteristic of a set.
But what does this intuitively mean? We have expressions of volumes of n-balls with radius of Bernoulli umbra (plus 1). What could it even mean intuitively?
P.S. Asked this question in Math.StackExchange several months ago, I think, I can now post it here as well.
