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15 votes
4 answers
3k 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 ...
Amritanshu Prasad's user avatar
14 votes
7 answers
3k views

A special type of generating function for Fibonacci

Notation. Let $[x^n]G(x)$ be the coefficient of $x^n$ in the Taylor series of $G(x)$. Consider the sequence of central binomial coefficients $\binom{2n}n$. Then there two ways to recover them: $$\...
T. Amdeberhan's user avatar
12 votes
2 answers
716 views

Alternating sum of hook lengths: Part I

Given $\lambda$ an integer partition of $n$, let $h_{ij}(\lambda)$ denote the hook length of cell $(i,j)$ in the Young diagram of $\lambda$. Is there a closed formula or a generating function for the ...
T. Amdeberhan's user avatar
11 votes
2 answers
1k views

Proofs of some combinatorial identities

Just wondering if anyone knows any references in the literature to bijections corresponding to the following simple generating function identities. Let $B(z)=\dfrac{1}{\sqrt{1-4z}}$ and $C(z)=\dfrac{1-...
Alexander Burstein's user avatar
8 votes
3 answers
536 views

Looking for a "cute" justification for a Catalan-type generating function

The Catalan numbers $C_n=\frac1{n+1}\binom{2n}n$ have the generating function $$c(x)=\frac{1-\sqrt{1-4x}}{2x}.$$ Let $a\in\mathbb{R}^+$. It seems that the following holds true $$\frac{c(x)^a}{\sqrt{1-...
T. Amdeberhan's user avatar
4 votes
0 answers
208 views

Extract this constant term

Given a Laurent polynomial $F$ in the variables $\mathbf{t}=(t_1,\dots,t_n)$, let $CT_{\vec{\mathbf{t}}}\,F$ denote its constant term. For example, $CT_{t_1,t_2}((8t_1-\frac1{3t_1t_2})(5t_1t_2+t_2^2+\...
T. Amdeberhan's user avatar
1 vote
1 answer
344 views

Products involving exponents of tribonacci numbers

The Fibonacci numbers $F_n$ can be given by $$\sum_{k\geq0}F_kx^k=\frac{x}{1-x-x^2}.$$ Among many many properties of this sequence, consider the following two results: (1) the coefficients of the ...
T. Amdeberhan's user avatar