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comment Showing ${n + a - 1 \choose a - 1} = \sum_{k = 0}^{\left\lfloor n/2 \right\rfloor} {a \choose n-2k}{k+a-1 \choose a-1}$
Thank you for pointing this out. I do sometimes use additional properties of complex variables like for example in this MSE proof of an identity by Gosper.
1d
answered Permutations minus Transpositions
1d
comment probability of $k$ boxes contain exactly $1$ ball
(+1). I had tried inclusion-exclusion but for some reason couldn't get it to work.
1d
awarded  Revival
2d
revised probability of $k$ boxes contain exactly $1$ ball
example
2d
answered probability of $k$ boxes contain exactly $1$ ball
Jul
27
revised $\int_0^\infty \frac{\log(x)}{x^2+\alpha^2}$ using residues
real vs imaginary
Jul
27
revised $\int_0^\infty \frac{\log(x)}{x^2+\alpha^2}$ using residues
language
Jul
27
revised $\int_0^\infty \frac{\log(x)}{x^2+\alpha^2}$ using residues
bonus
Jul
27
answered $\int_0^\infty \frac{\log(x)}{x^2+\alpha^2}$ using residues
Jul
26
revised How to find the number of all the possible ordered trees with n edges and k leaves?
verify catalan numbers
Jul
26
comment Given 5 colors to choose from, how many ways can we color the four unit squares of a $2\times 2$ board
Hint: there are four rotations in the group $C_4$ acting on the squares. Factor them into cycles. For Burnside we need to count the colorings that are fixed by each permutation and average those counts. A coloring is fixed if it is constant on each cycle. How many admissible colorings does that produce for the four different permutations given that we have five available colors?
Jul
26
awarded  complex-analysis
Jul
26
revised How to find the number of all the possible ordered trees with n edges and k leaves?
Maple
Jul
26
revised How to find the number of all the possible ordered trees with n edges and k leaves?
internal nodes
Jul
26
revised How to find the number of all the possible ordered trees with n edges and k leaves?
simplification
Jul
26
answered How to find the number of all the possible ordered trees with n edges and k leaves?
Jul
22
revised If I randomly generate a string of length N from an alphabet {A, B, C}, what's the likelihood that k characters will be the same OR BETTER?
remove first version
Jul
22
revised If I randomly generate a string of length N from an alphabet {A, B, C}, what's the likelihood that k characters will be the same OR BETTER?
second and third
Jul
22
revised If I randomly generate a string of length N from an alphabet {A, B, C}, what's the likelihood that k characters will be the same OR BETTER?
maple code