query: Closed form of $a_i = \lvert\{\, (e_1, \dotsc, e_r) \in [0, q)^r : e_1 + \dotsb + e_r = i \,\}\rvert $
positive_passages.docid: 149744
positive_passages.text: How can I determine the number of unique hands of size H for a given deck of cards?
negative_passages.docid: 146762
negative_passages.text: Closed form for $\sum_{k=0}^{n} \binom{n}{k}\frac{(-1)^k}{(k+1)^2}$
negative_passages.docid: 35345
negative_passages.text: How do i prove invertibility of this linear transformation
negative_passages.docid: 1416
negative_passages.text: closed form of $\int_{0}^{2\pi}\frac{dx}{(a^2\cos^2x+b^2\sin^2x)^n}$
negative_passages.docid: 141909
negative_passages.text: Prove ${2}^{n}\cdot(2m+1)-1 $ is invertible
negative_passages.docid: 18585
negative_passages.text: Showing $A-I$ is invertible, when $A$ is a skew-symmetric matrix
negative_passages.docid: 46106
negative_passages.text: Prove that $D_r \circ D_s = \{ (x,y) \in \mathbb{R}^2 \ | \ |x-y|<r+s \}$, where $D_a = \{ (x,y) \in \mathbb{R}^2\ | \ |x-y| < a \}$
negative_passages.docid: 115583
negative_passages.text: prove that $a^2(p-q)(p-r)+b^2(q-p)(q-r)+c^2(r-p)(r-q)\ge 0$
negative_passages.docid: 18477
negative_passages.text: $(W_1+W_2+\cdots+W_n)^a \leq W_1^a +\cdots + W_n^a$ for $n$ integer, $n\geq 2$, $W\gt 0$ and $a$ constant, real, $0\lt a\lt 1$
negative_passages.docid: 119606
negative_passages.text: Closed form of summation
negative_passages.docid: 22077
negative_passages.text: Closed form for the sequence defined by $a_0=1$ and $a_{n+1} = a_n + a_n^{-1}$
negative_passages.docid: 49741
negative_passages.text: Finding a closed form for a recurrence relation $a_n=3a_{n-1}+4a_{n-2}$
negative_passages.docid: 153612
negative_passages.text: Finding $g_i:\mathbb{R}^n\to\mathbb R$ s.t $f(x)=\sum\limits_{i=1}^nx_i\cdot g_i(x)$
negative_passages.docid: 108740
negative_passages.text: $C^1[a,b]$ is closed in $C[a,b]$
negative_passages.docid: 54735
negative_passages.text: What is the closed form of $\int e^{x^2} \, dx$?
negative_passages.docid: 78008
negative_passages.text: A conjectured closed form of $\int\limits_0^\infty\frac{x-1}{\sqrt{2^x-1}\ \ln\left(2^x-1\right)}dx$
negative_passages.docid: 145900
negative_passages.text: Proof of a double summation reduction identity