Let $F(n,\ell)$ be the matrix with coefficients $$F_{i,j}(n,\ell)=[t^{\ell j-i}] \left(\frac{1-t^\ell}{1-t}\right)^n,\,\;\;\;1\leq i,j \leq n$$ Above Pat Devlin pointed out that it suffices to show that the $(n-1)\times (n-1)$ submatrix $L(n,\ell)$ with coefficients $$L_{i,j}(n,\ell)=[t^{\ell j-i}] \left(\frac{1-t^\ell}{1-t}\right)^n,\,\;\;\;1\leq i,j \leq n-1$$ has eigenvalues $\ell,\ell^2,\ldots,\ell^{n-1}$.

In fact, for positive integer $\ell\geq 2$ the matrices $P(n,\ell)$ with coefficients $$P_{i,j}(n,\ell)=\frac{1}{\ell^n} [t^{(j+1)\ell-i-1}] \left(\frac{1-t^\ell}{1-t}\right)^{n+1},\,\;\;\; 0\leq i,j \leq n-1$$ are known. They are the transition matrices for the Markov chains describing the propagation of carries when $n$ integers which have independent uniform $\ell$-ary ''digits'' are added (clearly $\ell^n\cdot P(n,\ell)=L(n+1,\ell))$.

These matrices are subject of the fascinating article Carries, Combinatorics and an Amazing Matrix by John Holte (American Mathematical Monthly, 104 (2), 1997)).

Holte proved that $P(n,\ell)$ has eigenvalues $1,\ell^{-1},\ldots,\ell^{-(n-1)}$, that the eigenvectors do not depend on the base $\ell$, and described the left and right eigenvectors explicitly. He also showed that $P(n,a)\cdot P(n,b)=P(n,ab)$.

The $P(n,b)$ also appear in the probability of card shuffling. They are the transition matrices for the Markov chains describing the descents in the permutations generated by shuffling a deck of $n$ cards with successive $b$-shuffles. (Persi Diaconis and Jason Fulman, Carries, shuffling and an amazing matrix, AMM November 2009).

Let $F(n,\ell)$ be the matrix with coefficients $$F_{i,j}(n,\ell)=[t^{\ell j-i}] \left(\frac{1-t^\ell}{1-t}\right)^n,\,\;\;\;1\leq i,j \leq n$$ Above Pat Devlin pointed out that it suffices to show that the $(n-1)\times (n-1)$ submatrix $L(n,\ell)$ with coefficients $$L_{i,j}(n,\ell)=[t^{\ell j-i}] \left(\frac{1-t^\ell}{1-t}\right)^n,\,\;\;\;1\leq i,j \leq n-1$$ has eigenvalues $\ell,\ell^2,\ldots,\ell^{n-1}$.

In fact, for positive integer $\ell\geq 2$ the matrices $P(n,\ell)$ with coefficients $$P_{i,j}(n,\ell)=\frac{1}{\ell^n} [t^{(j+1)\ell-i-1}] \left(\frac{1-t^\ell}{1-t}\right)^{n+1},\,\;\;\; 0\leq i,j \leq n-1$$ are known. They are the transition matrices for the Markov chains describing the propagation of carries when $n$ integers which have independent uniform $\ell$-ary ''digits'' are added (clearly $\ell^n\cdot P(n,\ell)=L(n+1,\ell)$).

These matrices are subject of the fascinating article Carries, Combinatorics and an Amazing Matrix by John Holte (American Mathematical Monthly, 104 (2), 1997)).

Holte proved that $P(n,\ell)$ has eigenvalues $1,\ell^{-1},\ldots,\ell^{-(n-1)}$, that the eigenvectors do not depend on the base $\ell$, and described the left and right eigenvectors explicitly. He also showed that $P(n,a)\cdot P(n,b)=P(n,ab)$.

The $P(n,b)$ also appear in the probability of card shuffling. They are the transition matrices for the Markov chains describing the descents in the permutations generated by shuffling a deck of $n$ cards with successive $b$-shuffles. (Persi Diaconis and Jason Fulman, Carries, shuffling and an amazing matrix, AMM November 2009 (arXiv preprint)).