I'm looking for the name of the following polynomials:
$\mathrm{p}_1 = 1$
$\mathrm{p}_2 = x - \frac{1}{2}$
$\mathrm{p}_3 = \frac{1}{2} x^{2} - \frac{1}{2}x +\frac{1}{6}$
$\mathrm{p}_4 = \frac{1}{6} x^{3} - \frac{1}{4}x^{2} + \frac{1}{6} x - \frac{1}{24}$
$\vdots$
These polynomials are the solutions to the integrals
$
\mathrm{p}_{n} =
\int\limits_{y_{1} = 0}^{1} \;
\int\limits_{y_{2} = 0}^{x - y_{1}}
\int\limits_{y_{3} = 0}^{x - y_{1} - y_{2}}
\cdots
\int\limits_{y_{n} = 0}^{x - \sum\limits_{i = 1}^{n - 1} y_{i}}
1\; \mathrm{d}y_{1} \cdots \mathrm{d}y_{n}
$
over an n-dimensional simplex with a constraint $x = \sum\limits_{i=1}^{n} y_{i}$.
They seem to obey the following recurrence relations:
$\frac{\mathrm{d} \mathrm{p}_{n}}{\mathrm{d} x} = \mathrm{p}_{n - 1}$
$\mathrm{p}_{n} = \int \mathrm{p}_{n - 1} \mathrm{d} x
+ \left( -1 \right)^{n - 1} \frac{1}{n!}$
So far, I found out that my polynomials look similar to Bernoulli polynomials and that the Appell sequences to which the Bernoulli polynomials belong, obey a similar derivative recursion relation $\frac{\mathrm{d} \mathrm{p}_{n}}{\mathrm{d} x} = n\mathrm{p}_{n - 1}$
edit:
Pedro Tamaroff pointed out that multiplying each polynomial by n! gives the binomial coefficients. Thus the polynomials could be written as:
$\mathrm{p}_{n} = \frac{1}{n!} \sum\limits_{i=1}^{n}\left(-1\right)^{n}\binom{n}{i} x^{n-i}$
Does anybody know something about these polynomials and do they have a special name? Thanks!
Note your polynomials upon multiplication by $n!$ are $$1\\2x-1\\3x^2-3x+1\\4x^3-6x^2+4x-1\\\cdots$$ The next one in the list is $5x^4-10x^3+10x^2-5x+1$. The coefficients are the binomial coefficients. $$1\\1,1\\1,2,1\\1,3,3,1\\1,4,6,4,1\\$$