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### Deriving global probabilities from local dynamics

I am interested in growth dynamics and, in particular, how to derive difference/differential/stochastic equations from local behavior of a system. I tried posting first on math.stackexchange (https://...
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### Reccurence relation for [closed]

Hi guys I am a little confused on a question I am working on. "Write the recurrence relation for: 5, 0, -8, -17, -25, -30,…" I am getting Cn= {5 if n=1} {Cn-1+(2n-1) if n > 1} ...
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### 2D sequence of integers [closed]

I found the following sequence of integers $a_n^k$, where $n$ and $k$ to extend to infinity. Each row are coefficients of a polynomial, similar to the coefficients of the Legendre polynomials. The ...
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### Arithmetical properties of certain recurrence relations

Consider the following recurrence relation: $$(4i+6j+1)(2i+3j)a_{i,j}=3j a_{i+1,j-1}-2i a_{i-1,j}+32i(i-1) a_{i-2,j+1}, i\ge0, j\ge0,$$ $$a_{0,0}=1.$$ This equation appeared in the article (...
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### Solving Recursive Expression for Counting Unrooted Tree Topologies

The book Bayesian Evolutionary Analysis with BEAST by Alexei J. Drummond et al. (2015) states at section 2.2.1 that the number of rooted unlabelled (binary) tree topologies $a_n$ is given by the ...
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I have this recursive equation: \begin{align*} F(m,n)=F&=F(m,n-1)+F(m-1,n)-F(m-1,n-1-m) \\\\ F(m,0)&=F\left(m,m(m+1)/2\right)=1\\ F(m,i)=0&=0\text{ if i<0, i> }i<0\text{ or }i&... 0answers 133 views ### Occasional'' Laurent phenomenon This question is motivated by Richard Stanley's A question on the Laurent phenomenon (motivated by his answer to the question what is the probability that a scissor became the champion?). He asked ... 0answers 200 views ### Degenerate linear recurrence sequences Let (u_n)_{n \geq 0} be a linear recurrence given byu_n = a_1 u_{n-1} + \cdots + a_k u_{n-k} \quad \forall n \geq k ,$$where u_0, \ldots, u_{k-1}, a_1, \ldots, a_k \in \mathbb{Z}. We recall ... 1answer 80 views ### Stationary distribution of random walk alias solving uncountably many linear equations [closed] Let us have interval I = (i_1,i_2), function f_1 : I \mapsto I, function f_2 : I \mapsto I. Let x_0, x_1, x_2, ... be series of random variables from interval I denoting random walk. ... 0answers 155 views ### Number of digits in n-th term of generalized Fibonacci/Narayana sequence Let d be a nonnegative integer, and let the sequence {F_d(n)} be defined as follows: F_d(n) = 1 for n = 0, 1, \ldots, d F_d(n) = F_d(n-1) + F_d(n-1-d) for n>d For d=0 the sequence ... 2answers 118 views ### Transforming a recurrence to the product of two other recurrences This question deals with sequence:$$a_0 = 0,\ a_1 = 7, \quad a_{n+2} = 14a_{n+1} - a_n + 6$$The question is about establishing that a_n is a composite number (except some finite cases). In one ... 0answers 134 views ### Help solving a recurrence relation For solving a related probability problem, I need to solve the following recurrence relation:- q(r,b,L)=\frac{b}{r+b}q(r,b-1,L)+\sum_{k=1}^{L}\frac{r}{r+b}\times\cdots \times \frac{r-(k-1)}{r+b-(k-1)... 1answer 296 views ### Polynomial recurrence relation covering the integers (and then Gaussian integers) Say that a polynomial recurrence relation (my terminology) for f_i is: k initial conditions setting f_1,\ldots,f_k to integers (\in \mathbb{Z}). A recurrence equation of the form f_i = a ... 0answers 76 views ### For which recurrence relations is it decidable whether a formal power series has a maximal zero coefficient? In this MSE question, I asked whether one can prove that a generating function has infinitely many coefficients equal to zero. The answer given (and accepted) to that rather broad question was “No”. ... 3answers 325 views ### Limiting probabilities for two-player game drawing random uniform numbers Consider this simple 2-person game I just made up: Player A goes gets to draw a uniform U[0,1] number up to X times. At any time, he may either keep his number, or draw a brand new uniform number. ... 2answers 281 views ### Strong divisibility of Lucas sequences Let a and b be relatively prime integers and let u_n be their associate Lucas sequence, i.e., the second order linear recurrence sequence satisfying u_0 = 0, u_1 = 1 and u_{n+2} = au_{n+1} +... 0answers 52 views ### A question on bivariate recurrence Is there a way to get a closed form of this recurrence (albeit approximate):$$A(n ,k ) = {1 \over {d^k}}A(n-1, k-1) + (1 - {1 \over {d^k}})A(n-1,k)$$Where, n,k \ge 0 and d \ge 2 are integers. ... 2answers 320 views ### Faster formula to compute sum over partitions Let f be a function from the positive integers to the real numbers (or some ring...). Let$$(\star) \quad F(n) = \sum_{n_1 \leq \cdots \leq n_j\atop n_1 + \cdots + n_j = n} f(n_1) \cdots f(n_j),  ...
While dealing with a problem related to intersection of hyperplanes I have come across the following recurrence to obtain the values of $K_{j}$ \begin{array}{cccccccccc} 1 & = & K_{1}\tbinom{...