Generating Functions

On March 22nd 2018, my number of the day was 25190 described as a member of OEIS A014334 whose terms arise from the exponential convolution of Fibonacci numbers with themselves. This meant the term arose from a generating function of a particular type, namely exponential. The term convolution means that the generating function is multiplied by itself to generate a new function. The first eleven terms of OEIS A014334 are 0, 0, 2, 6, 22, 70, 230, 742, 2406, 7782 and 25190. So how are these terms generated?

Firstly, we need to establish what the exponential generating function is for the Fibonacci numbers. It turns out to be: $$ \sum_{n=0}^{\infty} F_n \frac{t^n}{n!}$$ $$\text{where } F_n=\frac{\alpha^n-\beta^n}{\alpha-\beta} \text{ and } \alpha=\frac{1+\sqrt{5}}{2} \text{ and } \beta=\frac{1-\sqrt{5}}{2}$$
Now the convolution of the generating function with itself yields new terms that can be found from the original Fibonacci numbers. Let's define two exponential generating functions: $$ A(t)=\sum_{n=0}^{\infty} a_n \frac{t^n}{n!} \text{ and } B(t)=\sum_{n=0}^{\infty} b_n \frac{t^n}{n!}$$ $$ \text{then } A(t).B(t)=\sum_{n=0}^{\infty} \bigg(  \sum_{k=0}^{n} \binom{n}{k}.a_k.b_{n-k} \bigg) \frac{t^n}{n!}$$ Let's remember that \(a_n = b_n =F_n \) and so all the coefficients of this generating function can be found by setting \(n=0, 1, 2, 3, ... \). For example, when \( n=4 \) the coefficient is $$ \binom{4}{0}.F_0.F_4+\binom{4}{1}.F_1.F_3+\binom{4}{2}.F_2.F_2+\binom{4}{3}.F_3.F_1+\binom{4}{4}.F_4.F_0$$Remember that \( F_0=0, F_1=1, F_2=1, F_3=2, F_4=3 \) and so substituting these values, the coefficient becomes \( 1*0*3+4*1*2+6*1*1+4*2*1+1*3*0=22 \). The magnitude of these coefficients rises quickly as sequence OEIS A014334 shows. So this is how the number 25190 can be calculated (by setting n=10).

REFERENCE: https://www.mathstat.dal.ca/FQ/Scanned/11-3/church.pdf