I have a combinatorics problem and I've reduced it to finding coefficient that stands with $x^n$ in this polynomial,
$$(1+x)(1+x+x^2)...(1+x+x^2+...+x^n)$$
But now I'm stuck. Can someone help me please?
I have a combinatorics problem and I've reduced it to finding coefficient that stands with $x^n$ in this polynomial,
$$(1+x)(1+x+x^2)...(1+x+x^2+...+x^n)$$
But now I'm stuck. Can someone help me please?
What you are looking for is a row in the table of Mahonian numbers.
Here they are: Sloane's A008302.
That link gives you lots and lots of other links and references, and it includes a recurrence for calculating them (which I added a couple of years ago). Here, $T(1,)$ represents $1$ and $T(2,)$ represents $(1+x)$, etc.
$$T(1, 1) = 1,\qquad T(1, k\neq 1) = 0,\\ T(n, k) = T(n, k-1) + T(n-1, k) - T(n-1, k-n),\\ \text{or }T(n, k) = \sum_{j=0}^{n-1}\ T(n-1, k-j),\\$$
This product gives the Poincare series for the symmetric group $S_{n+1}$ considered as a Coxeter group of type $A_n$. The answer is therefore the number of permutations in $S_{n+1}$ that can be minimally expressed as a product of $n$ adjacent transpositions. I think this is likely the best formula you will find for this number.