I have been inspired by this question to think about the polynomial sequence $a : \mathbb{N} \rightarrow \mathbb{Q}[x]$ defined by $$a_n(x) = \int_0^{1{-}x}dy\,a_{n{-}1}(y),\qquad a_0 = 1.$$ The first few terms in the sequence are given by $$\begin{align*}a_0 &= 1\\ a_1 &= 1 - x\\ a_2 &= \frac12 (1 - x^2)\\a_3 &= \frac{1}{3!} (2 - 3 x^2 + x^3)\\a_4 &= \frac{1}{4!} (5 - 6 x^2 + x^4)\\a_5 &= \frac{1}{5!} (16 - 20 x^2 + 5 x^4 - x^5)\end{align*}.$$
So, here are two questions about this sequence.
- Can this sequence be written in terms of standard polynomial sequences? (eg. Legendre, Chebyshev etc.)
- Is there a way to find $a_n(0)$, either as an explicit function of $n$, or as an algebraic recursion relation in $a_{n-1}(0)$? (I think this would answer the original question nicely)
I generated the first 6 terms in Mathematica using With[{n = 6}, #[x] & /@ NestList[Function[{x}, Integrate[#[y], {y, 0, 1 - x}]]] &, 1 &, n] // Together, in case anyone is interested