An explicit formula for Bernoulli numbers in terms of Stirling numbers of the second kind
Feng Qi
TL;DR
The paper addresses the problem of expressing Bernoulli numbers $B_n$ explicitly via Stirling numbers of the second kind $S(n,k)$. It uses the Faà di Bruno formula and Bell polynomials to relate the derivatives of the Bernoulli generating function $\frac{x}{e^x-1}$ to combinatorial quantities, yielding explicit representations such as $\textup{B}_{n,k}(0,\overbrace{1,\dots,1}^{n-k})=\sum_{i=0}^k(-1)^i\binom{n}{i}S(n-i,k-i)$ and $B_n=\sum_{i=0}^n(-1)^i\frac{\binom{n+1}{i+1}}{\binom{n+i}{i}}S(n+i,i)$. It also shows that $\textup{B}_{n,k}(\tfrac{1}{2},\tfrac{1}{3},\dots)=\frac{n!}{(n+k)!}\textup{B}_{n+k,k}(0,\overbrace{1,\dots,1}^{n})$, enabling an alternate representation in terms of $S(n+i,i)$. The results connect to prior explicit formulas such as $B_n=\sum_{k=1}^n(-1)^k\frac{k!}{k+1}S(n,k)$ and provide a combinatorial route to compute Bernoulli numbers from Stirling numbers.
Abstract
In the note, the author discovers an explicit formula for computing Bernoulli numbers in terms of Stirling numbers of the second kind.