<s>[INST] Determining the coefficients of $(1 + x + x^2 +\cdots+x^n)^{n-1}$ Suppose we have the following polynomials:
$$f_1(x)=(1 + x + x^2)$$
$$f_2(x)=(1 + x + x^2 + x^3)^2$$
$$f_3(x)=(1 + x + x^2 + x^3 + x^4)^3$$
$$f_4(x)=(1 + x + x^2 + x^3 + x^4 + x^5)^4$$
$$\vdots$$
$$f_{n-1}(x)=(1 + x + x^2 + x^3 +x^4+ x^5+\cdots+x^n)^{n-1}$$ 
upon expanding them we get:
$$f_1(x)=1 + x + x^2$$
$$f_2(x)=1 + 2 x + 3 x^2 + 4 x^3 + 3 x^4 + 2 x^5 + x^6$$
$$f_3(x)=1 + 3 x + 6 x^2 + 10 x^3 + 15 x^4 + 18 x^5 + 19 x^6 + 18 x^7 + 
 15 x^8 + 10 x^9 + 6 x^{10} + 3 x^{11} + x^{12}$$
$$f_4(x)=1 + 4 x + 10 x^2 + 20 x^3 + 35 x^4 + 56 x^5 + 80 x^6 + 104 x^7 + 
 125 x^8 + 140 x^9 + 146 x^{10} + 140 x^{11} + 125 x^{12} + 104 x^{13} + 
 80 x^{14} + 56 x^{15} + 35 x^{16} + 20 x^{17} + 10 x^{18} + 4 x^{19} + x^{20}$$
$$\vdots$$
$$f_{n-1}(x)=1 + ?x + ?x^2 + ?x^3 +?x^4+ ?x^5+\cdots+?x^{n(n-1)}$$ 
I'm wondering how to determine the coefficients for the n-th order? I can observe that the coefficients are symmetric. 
 [/INST] $\sum_{i=0}^{k\cdot m}c_ix^i=(1+x^1+x^2+...+x^m)^{k}$ is the generating function of the number of weak integer compositions (integer compositions with repetitions of $0$) of integer $i$ with $k$ parts where all parts are lower equal to $m$.
Unfortunately, it is not yet in OEIS.
$k,m>0$
Their coefficients are:
$$c_i=\sum_{j=0}^{\frac{i+k-1}{m+1}}(-1)^{j}\binom{k}{j}\binom{i+k-j(m+1)-1}{k-1}.$$
[Stanley 1999], Mistake in the closed form formula for the number of restricted compositions?
with $n>0$, $k=n$, $m=n+1$:
$$c_i=\sum_{j=0}^{\frac{i+n-1}{n+2}}(-1)^{j}\binom{n}{j}\binom{i+n-j(n+2)-1}{n-1}$$
$\ $
[Stanley 1999] Stanley, R. P.: Enumerative Combinatorics Vol. I. Cambridge University Press, 1999
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