If you are interested in approximations to your ratios, you may find the accepted answer (and some comments of mine) to this MathOverflow post useful: Sum of 'the first k' binomial coefficients for fixed …

Aug 17, 2015 · Zaslavsky's formula is a very important formula in enumerative combinatorics, as well as geometric combinatorics, and the basis for important developments in topological combinatorics.

In addition to the OP's 2011 paper with Ž. Jurić: A New Formula for the Number of Combinations of Permutations of Multisets Applied Mathematical Sciences, Vol. 5, 2011, no. 18, 875-881 there is a …

Feb 17, 2025 · How to prove that $$\\sum_{b=0}^{[\\frac{n}{2}]}(-1)^{n-b-1}(n-b-1)!\\binom{n}{n-2b}\\frac{\\binom{2b}{2}\\binom{2b-2}{2}\\dots \\binom{2}{2}}{b!}2^{n-b}$$ equals ...

First component (for i = 0) is just a regular combination with repetition, then I subtract all that have at least 1 box overfilled, then I need to add those that have at least 2 box overfilled (since earlier I …

0 I have been struggling with this problem and hope someone could help. I am trying a variation of non-repetitive combination scenario. I can use the formula n!/r!x (n-r)! to find non-repetitive combinations …

Jan 31, 2024 · Preliminary tests show that this series performs pretty faster than the fastest known series for such constant that is based on a 4-term Machin-like formula (a linear combination of …

asymptotic or approximate formula for a combination expression Ask Question Asked 12 years, 10 months ago Modified 12 years, 10 months ago

Oct 26, 2015 · Rather than expressing this as a linear combination with of 1, n,n2, ⋯,n2q 1, n, n 2,, n 2 q with rational coefficients, it is better to express it as a linear combination of (n0),(n1), ⋯,(n2q) (n 0), …

Sep 21, 2020 · The trace of a matrix is the sum of the eigenvalues and the determinant is the product of the eigenvalues. The fundamental theorem of symmetric polynomials says that we can write any …