In mathematics,   Faulhaber's formula,   named after Johann Faulhaber,   expresses the sum of the p-th powers of the first n positive integers as a (p + 1)th-degree polynomial function of n,   the coefficients involving Bernoulli numbers.



Task

Generate the first 10 closed-form expressions, starting with p = 0.



Related tasks

   Bernoulli numbers.
   evaluate binomial coefficients.


See also

   The Wikipedia entry:   Faulhaber's formula.
   The Wikipedia entry:   Bernoulli numbers.
   The Wikipedia entry:   binomial coefficients.


Bsecond=:verb define"0
  +/,(<:*(_1^[)*!*(y^~1+[)%1+])"0/~i.1x+y
)
 
Bfirst=: Bsecond - 1&=
 
Faul=:adverb define
  (0,|.(%m+1x) * (_1x&^ * !&(m+1) * Bfirst) i.1+m)&p.
)