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3
votes
0answers
39 views

Finite realization of irrational transfer functions

In the field of digital signal processing, linear time-invariant systems play a distinguished role. These are the systems for which there exists an impulse response, a function ...
3
votes
1answer
183 views

Generating function of alternating even terms in the Vandermonde Convolution

I have \begin{equation} G(x) = \sum_{i = 0}^{\infty} \sum_{r = 0}^{\infty} (-1)^i \binom{k}{2i} \binom{n-k}{r} x^{2i} x^{r} = \frac{1}{2} \left( (1 + x{\iota})^{k} + (1 - x \iota)^{k} \right)(1 + ...
11
votes
1answer
220 views

Generating function of a sequence is not algebraic

Let we have a sequence $\{a_{n}\}$, such that $\forall n \,\, a_{n}>0$ and $a_{n} \rightarrow\infty, n\rightarrow\infty$. Also let's suppose that we have a subsequence $\{a_{n_{k}}\}$ such that ...
6
votes
3answers
373 views

Tricky two-dimensional recurrence relation

I would like to obtain a closed form for the recurrence relation $$a_{0,0} = 1,~~~~a_{0,m+1} = 0\\a_{n+1,0} = 2 + \frac 1 2 \cdot(a_{n,0} + a_{n,1})\\a_{n+1,m+1} = \frac 1 2 \cdot (a_{n,m} + ...
55
votes
8answers
8k views

What is Lagrange Inversion good for?

I am planning an introductory combinatorics course (mixed grad-undergrad) and am trying to decide whether it is worth budgeting a day for Lagrange inversion. The reason I hesitate is that I know of ...
13
votes
3answers
1k views

A seemingly simple combinatorial object that must have an easy generating function

One more question related to my earlier "Special" meanders. I am trying to isolate simplest problems related to it. Here is one. For a composition (i. e. a tuple of natural numbers) ...
4
votes
1answer
217 views

Eliminating a variable from a two-variable linear recurrence

In attempting to enumerate a combinatorial class of objects, I've come to a bivariate recurrence: $$ a_{n,k} = 2a_{n,k-1} + (k+1)a_{n-1,k+1} - ka_{n-1,k} - a_{n,k-2} + a_{n-1,k-1}. $$ Together with ...
4
votes
2answers
1k views

Number of 1 in binary representation of n

Let $1(n)$ be the number of digits $1$ in binary representation of number $n$. For example, $13=1101_2$ so $1(13)=3\\$ Is there explicit form of $\,\,\sum{1(i)x^i} $? I checked OEIS and didn't find ...
3
votes
1answer
169 views

Solving recursion / finding generating function of a probability mass function

I am assessing the probability distribution on a running time of some algorithm that we've developed. I am looking for a family of probability mass functions $f_n$ with the following recurrence: $$ ...
11
votes
4answers
1k views

Ordinary Generating Function for Bell Numbers

In the OEIS entry for Bell numbers, there appears a generating function $$\sum_{k=0}^\infty B_k t^k = \sum_{r=0}^\infty \prod_{i=1}^r \frac{t}{1-it}$$ However, I could not locate any proof of ...
8
votes
1answer
523 views

Another formula for Bell numbers

Here is an observation (thanks to OEIS): $$\sum_{i=0}^\infty \frac{i^k}{i!}= B_k e,$$ where $B_k$ is the $k$-th Bell number. I might be having reading comprehension issues, but I don't see this ...
3
votes
0answers
158 views

Combination of Generating Functions

Suppose I have the following generating functions: $$\frac{x^ke^{\left(z-\frac{1}{N}\right)x}}{N^{k-1}k!\sum_{j=0}^{N-1}w_N^{-jk}e^{\frac{w_N^jx}{N}}}=\sum_{j=0}^\infty H_{N,k,j}(z)\frac{x^j}{j!}$$ ...
3
votes
5answers
664 views

Generating-functions: is there a relationship between a generating function and the corresponding squared generating function

Let's say we have a sequence $T(n)$ with the corresponding generating function $$A(t) = \sum_{n = 0}^\infty T(n) t^n$$ Is there some relationship between the two functions $A(t)$ and $A(t^2)$? And ...
2
votes
0answers
168 views

Hilbert series of the weight 0 sub-algebra of the algebra of functions on GL(N)

Let $A$ be the algebra of polynomials in $N^2$ variables $x^i_j$, $i,j=1,\dots,N$. It is $\mathbb{Z}^N$ graded, with $\text{weight}(x^i_j)=e_i-e_j$. Here $(e_1,\dots,e_n)$ is the standard basis of ...
0
votes
0answers
91 views

Multidimensional recurrence relations

There are many methods of solving one-dimensional homogeneous linear recurrence relations, i.e. such of the form $$ a_n = \sum_{k=1}^{m}\alpha_ka_{n-k}.$$ Most widespread use linear algebra or ...
23
votes
3answers
3k views

Is the sum $\sum\limits_{j=0}^{k-1}(-1)^{j+1}(k-j)^{2k-2} \binom{2k+1}{j} \ge 0?$

I am trying to prove $\sum\limits_{j=0}^{k-1}(-1)^{j+1}(k-j)^{2k-2} \binom{2k+1}{j} \ge 0$. This inequality has been verified by computer for $k\le40$. Some clues that might work (kindly provided by ...
2
votes
0answers
69 views

Inverses of probability generating functions: positivity of derivatives

Let $\mathcal{G}$ be the set of probability generating functions of random variables taking positive integer values, considered as functions on $[0,1]$. So $G\in\mathcal{G}$ can be written ...
1
vote
3answers
183 views

Hypergeometric sum specific value

How to show? $${}_2F_1(1,1;\frac{1}{2}, \frac{1}{2}) = 2 + \frac{\pi}{2} $$ It numerically is very close, came up when evaluating: $$ \frac{1}{1} + \frac{1 \times 2}{1 \times 3} + \frac{1 \times 2 ...
0
votes
0answers
55 views

Direct proof that a certain generating function is D-finite

Consider the set $T^{2,3}_n$ of all non-planar rooted trees with $n$ leaves labelled by $1,2,\ldots,n$ where each internal vertex can have two or three children. If we think of the binary / ternary ...
2
votes
2answers
391 views

Closed formula for the generating function of the sequence of powers

Does anyone know of a closed formula for the function $f_k(x)=\sum_{n=1}^{\infty}{n^k x^n}$ ? That is, the generating function of the sequence $1^k,2^k,3^k...$. It is not hard to see that ...
4
votes
1answer
103 views

Calculation of one constant similar to MZV

The series arose in the calculation of Mean value of a function associated with continued fractions: $$C=\sum_{1\le b\le d<\infty}\frac{1}{b(b+d)d^2}.$$ Obviously $C=C_1-C_2,$ where ...
14
votes
2answers
234 views

Generating functions for objects with irrational sizes

A problem I'm investigating concerns a combinatorial class in which the 'atoms' have irrational sizes. It seems likely that this is something that has been considered before, but I haven't been able ...
16
votes
2answers
624 views

Combinatorial meaning of the functional equation for logarithm

If we set $\exp(x)=\sum x^k/k!$, then $\exp(x+y)=\exp(x)\cdot \exp(y)$. In terms of coefficients it means that $(x+y)^n=\sum \frac{n!}{k!(n-k)!} x^ky^{n-k}$, i.e. just binomial expansion. Now ...
9
votes
2answers
263 views

Asymptotic growth rate of coefficients of generating function

how to calculate the asymptotic growth rate of coefficients generating function $T(z)$ satisfied this identity $T(z)=z+\frac{T(z)^3}{6}+\frac{T(z^2)T(z)}{2}+\frac{T(z^3)}{3}$
0
votes
1answer
79 views

Solving the difference equation $h(\vec x)\cdot A(\vec x)=\sum_{i=1}^m A(\vec x - \vec e_i)$

I am trying to solve the following difference equation: $$h(x_1,x_2,x_3,x_4) \cdot A(x_1,x_2,x_3,x_4)=A(x_1-1,x_2,x_3,x_4)+A(x_1,x_2-1,x_3,x_4)+A(x_1,x_2,x_3-1,x_4)+A(x_1,x_2,x_3,x_4-1)$$ with ...
4
votes
1answer
275 views

Generating function for number of different tessellation checkered rectangle

Let $R_n$ be checkered rectangle sized $n \times 4, n \ge 1$. Let $a_n$ be number of different $R_n$ tiling with rectangles sized $1 \times 3$. $\ \ \ $ $\ \ \ $ $\ \ \ $ $\ \ \ $ $\ \ \ $ $\ \ \ ...
4
votes
0answers
157 views

Puzzling behaviour of a recursively defined sequence of functions

This question arose in connection with another problem that I described earlier in Constructing a generating function using a series with all negative and positive powers of a variable. I had certain ...
2
votes
2answers
357 views

Is there a nice generating function proof of the following identity?

Consider the Jordan function $J_2(n)$ defined by $$ J_2(n) = \#\{x \in (\mathbb{Z}/n)^2 \mid ord(x) = n\} $$ (this is OEIS A007434). One can prove the following identity pretty easily: $$ \sum_{d \mid ...
6
votes
3answers
484 views

A diagonal operation on power series

Given a formal power series $f(x,y)=\sum_{n,m\ge 0}f(n,m)\: x^n y^m$ in two variables $x$ and $y$ over some field of characteristic zero, e.g. the field of complex numbers $\mathbb C$, we define a new ...
7
votes
1answer
199 views

Constructing a generating function using a series with all negative and positive powers of a variable

Trying to count certain combinatorial structures, I arrived at a construction of their generating function through a very inconvenient procedure. I realize that anybody who will read this has right ...
5
votes
0answers
181 views

Asymptotics and combinatorics

Wright's expansion of $$ (1-z)^b\exp[A/(1-z)^c], \text{for } A > 0, 0<c<1\tag{1} $$ is, in the words of the late, great Mark Kac "well known to those that know it well". (See, for example, ...
12
votes
1answer
309 views

Generating function of the Thue-Morse sequence

Let $T$ be the generating function of the Thue-Morse sequence; thus, $T(x)=x+x^2+x^4+x^7+\dotsb$. It is known that $T$ satisfies the nice congruence $$ (1+x)^3 T^2(x) + (1+x)^2 T(x) + x \equiv 0 ...
44
votes
3answers
5k views

Hirzebruch's motivation of the Todd class

In Prospects in Mathematics (AM-70), Hirzebruch gives a nice discussion of why the formal power series $f(x) = 1 + b_1 x + b_2 x^2 + \dots$ defining the Todd class must be what it is. In particular, ...
3
votes
1answer
421 views

p-adic poly-Bernoulli numbers

We can define p-adic Bernoulli polynomials by using q-integral on $\mathbb{Z}_p$ and Taekyun Kim's method. But how can we define p-adic poly-Bernoulli numbers and polynomials by using integral on ...
2
votes
1answer
633 views

Probability generating function zero implies random variable is infinite

Let $V$ be a random variable supported on the nonnegative integers (including $\infty$) and $f(x) = \mathbf E x^V$ be the probability generating function. In our model $V$ is the number of visits to ...
3
votes
0answers
126 views

Generating function of a sequence involving reciprocals of binomial coefficients

Question: Is there a closed-form expression for the following sum $$ F(z,k,r)=\sum_{n=0}^{r} \frac{z^n}{{n+k} \choose {k}}\label{sum}\tag{1} $$ where $z\in\mathbb{C}$, and $r$, $k$ are non-negative ...
5
votes
3answers
1k views

Asymptotics of a hypergeometric series/Taylor series coefficient.

I was planning on figuring this problem out for myself, but I also wanted to try out mathoverflow. Here goes: I wanted to know the asymptotics of the sum of the absolute values of the Fourier-Walsh ...
8
votes
2answers
330 views

Determining the Lambert series for $xq+x^2q^4+x^3q^9+…+x^nq^{n^2}+…$

I am trying to determine the polynomials $P_n(x)$ from $$ xq+x^2q^4+x^3q^9+...+\ x^nq^{n^2}+...=\sum_{n\geqslant1}\frac{P_n(x)q^n}{1-xq^n}; $$ that is, $$ \sum_{d|n}x^{\frac ...
4
votes
0answers
71 views

Is there a nice way to invert this expression?

Let us first define the Euler polynomials to be the polynomials $P_n(q)$ that satisfy $$ \frac{qP_n(q)}{(1 - q)^{n+1}} = \Big(q\frac{d}{dq}\Big)^n\frac{q}{1 - q}. $$ For example, $P_0(q) = P_1(q) = ...
3
votes
0answers
126 views

Involutions on $[0,1]$ given by power series (related to probability generating functions)

Let $A$ be a function from $[0,1]$ to $[0,1]$. $A$ is an involution if $A(A(x))=x$ for all $x\in[0,1]$. Which involutions $A$ exist such that $A(x)=\sum_{k=0}^\infty a_k x^k$ with $a_0=1$ and ...
1
vote
1answer
64 views

Restricted partitions with square terms only

Let $P(N,M,n)$ be the number of partitions of $n$ such that each term is $\le N$ and there are at most $M$ terms. So we know the generating function for $P(N,M,n)$ is $ \frac{(q)_{N+M}}{(q)_M ...
3
votes
1answer
386 views

A particularly “natural” algebraic structure with three commutative, pairwise-distributive operations

EDIT: As mentioned in my answer below, I was mistaken in thinking Dirichlet convolution distributes over ordinary convolution. I'm leaving this question here for reference. I keep stumbling on the ...
2
votes
1answer
186 views

Any interesting properties of the matrix $M:=(m_{ij})$ with $m_{ij}=min(i,j)$? [closed]

Do you know any interesting properties on the matrix $M(n):=(m_{ij})$ of size $n \times n,$ where $m_{ij}= \text{min}(i,j)$ ? The matrix $M$ enumerates certain combinatorial objects.
6
votes
1answer
121 views

Asymptotics of a Bivariate Generating Function

I have the following generating function, $$G(x,y)=\sum_{n,k \geq 0}a(n,k)x^ny^k = \frac{(y^2-y)x+1}{(y-y^3)x^2-(y+1)x+1}$$ and I am interested in obtaining an asymptotic for the sequence $a(n,k)$ ...
1
vote
1answer
159 views

How do powers affect asymptotics in generating functions?

Let $a_n$ be a sequence of non-negative real numbers, and $A(x) = \sum_{n=0}^{\infty} a_n \frac{x^n}{n!}$ its exponential generating function. Also, suppose $B(x) = \sum_{n=0}^{\infty} b_n ...
12
votes
2answers
401 views

Asymptotics of coefficients of implicitely defined generating function

I have two integer sequences $\{a_n\}_{n=0}^\infty$ and $\{b_n\}_{n=0}^\infty$. Explicit formulas for the $a_n$ are known and their asymptotic growth is fully understood. My wish is to also understand ...
10
votes
5answers
1k views

Use of everywhere divergent generating functions

Generating functions are well-known to be much useful in combinatorics. But, maybe just since I am illiteral, all the applications coming in mind deal with power series, which are not just formal, but ...
9
votes
0answers
112 views

types of generating functions (ordinary, exponential, ???) closed under substitution

A nice feature of ordinary and exponential generating functions is that they are closed under substitution: if $F(z)$ and $G(z)$ both have integer coefficients, then $F(G(z))$ also has integer ...
3
votes
0answers
59 views

Using Mellin transform for a certain function

In short, I want to use the Mellin transform to obtain the asymptotic behavior of the sequence $D_n = \frac{ [z^n] D(z)} {C_n}$ where $$ D(z) = \frac 1{2z}\sum_{p \ge 1}C_p \left( ...
5
votes
1answer
370 views

Repertory of the different sorts of operads

Many different types of operads have emerged in recent years (symmetric, shuffle, cyclic, anticyclic, coloured, etc.). I would like, for any of these, list the following data: Description of the ...