A fairly direct consequence of the indefinite sum m Sum Pochhammer(k,p) = Pochhammer(m,p+1)/(p+1), k=1
which is in turn a direct consequence of the rule of formation of Pascal's triangle,
nCk + nC(k+1) = (n+1)C(k+1), is
bern(p) = -(-1)^j*(p+j)*sum((-1)^k*stirling_s1(k+1,j)*stirling_s2(p+j-1,k)/(k+1),k,0,p+j-1)/binom(p+j,j) p + j - 1 k ==== (- 1) s S j \ k + 1, j p + j - 1, k (- 1) (p + j) > ------------------------------ / k + 1 ==== k = 0 B = - -------------------------------------------------------, p>0. p binom(p + j, j)
For p:=1,
bern(p) = sum((-1)^k*k!*stirling_s2(p,k)/(k+1),k,0,p)
p k ==== (- 1) k! S \ p, k B = > ---------------. p / k + 1 ==== k = 0
This latter, at least, must be in G, Knuth, and P. --rwg
Much better: eqn (38) of http://mathworld.wolfram.com/BernoulliPolynomial.html is the generalization of the latter to Bernoulli polynomials.