# Questions tagged [generating-functions]

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### Conjectural congruences for numbers related to Littlewood-Richardson coefficients

For $n \geq 0$, let $a_n$ be the square of the Euclidean length of the vector of Littlewood-Richardson coefficients of $\sum_{\lambda \vdash n} s_\lambda^2$, where $s_\lambda$ are the symmetric Schur ...
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### Property of an integer sequence related to series reversion

Thinking of some questions of homotopical algebra for operads, I ended up with a following question, perhaps someone will recognize something here: Let $\{a_n\}_{n\ge 2}$ be a sequence of nonnegative ...
39 views

### String partition function

Hey I am a bit confused because when reading physics papers I encounter the free energy as a generating function of topological invariants as integrals over certain compactified moduli spaces. For ...
214 views

### Analytic expression for the coefficient of a multivariate polynomial

Does there exist some method for finding an analytic expression for the coefficient of $z_1^kz_2^kz_3^k$ in: $$[(1+z_1)(1+z_2)(1+z_3)(1+z_1z_2)(1+z_1z_3)(1+z_2z_3)(1+z_1z_2z_3)]^{k}$$ or is it ...
604 views

### Matrices of combinatorial sequences that are inverse in two ways

I'm interested in pairs $A=(a_{i,j})_{i,j=0,1,\ldots}$ and $B=(b_{i,j})_{i,j=0,1,\ldots}$ of infinite matrices for which: They are uni-lower-triangular, i.e., $a_{i,i}=1$ for all $i$ and $a_{i,j}=0$ ...
1 vote
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315 views

### How are Sheffer polynomials related to Lie theory?

Sheffer polynomials are orthogonal polynomial sets $\{P_n(x)\}$ having generating function $P(x,t) = \sum_{n=0}^{\infty}P_n(x)t^n=A(t)e^{xu(t)}$. This form reminds me of the Lie group–Lie algebra ...
1 vote
107 views

### Hankel determinants of the sequence $(\binom{n}{m})_{n\ge0}$ and related sequences

I posted (https://math.stackexchange.com/questions/4363151/generating-functions-of-hankel-determinants-related-to-hoggatt-triangles) this question on Mathematics StackExchange but have not received a ...
238 views

### Tanglegrams and functional equations of M. Somos

Recent references on the matter at hand include, a lecture slide The Konvalinka-Amdeberhan conjecture and plethystic inverses and a preprint on Counting tanglegrams with species by I. Gessel; the ...
167 views

1 vote
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### Log-concavity of sequence related to overpartitions

The number $p_1(n)$ of overpartitions of $n$ is generated by $$\sum_{n\geq0}p_1(n)\,q^n=\prod_{k=1}^{\infty}\frac{1+q^k}{1-q^k}.$$ Let $t\in\mathbb{N}$. Now, extend this to construct a family of ...
1 vote
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### Two-variable generating functions over coprime pairs

I am studying a sequence $(\alpha_{p,q})$ indexed by a pair of coprime integers; this sequence arises naturally in the study of a particular set of spaces in geometric topology, but unfortunately the ...
123 views