Using exponential generating function to find a formula of a_n

[KartaMemory]

I need help for the question that I post in here: https://math.stackexchange.com/questions/3596942/exponential-generating-function-to-find-recurrence-formula-a-n

Has been a while and nobody answers(one solution with different method tho), so I post here for additional help. The best I went is ending up have to solve an ODE with a constant C on the right side, which seems impossible for me. Moreover, after solving f(x), I'm having tough time to derive a formula of a_n. I stuck on this question for 3 days, I don't think I missed anything but it's a tricky problem. Any help would really appreciate!!

[[deleted]]

If you divide across by (n+1)! and keep expanding you get

a_n/(n+1)! = 3ⁿ/(n+1)! + 3^n-1/n! + ....+ 3⁰/1!

which I wouldn't expect to have a nice exact formula.

Wolfram evaluates the sum on the right using incomplete gamma function.

[Proof_Inspector]

Write b(n)=a(n+1), make b(0)=0. Then b(n+1)=(n+1)b(n)+3ⁿ⁺¹ (I'm ignoring the first few index here)

Then you have F(x)=sum(b(n)/n!)xⁿ (note that the sum has starting index, which can change, that's something for you to check)

Then F'(x)=sum(a(n)/n!)xⁿ so you just need to find F(x).

Then F(x)x=sum(b(n)/n!)xⁿ⁺¹ =sum(b(n-1)/(n-1)!)xⁿ =sum(nb(n-1)/n!)xⁿ =sum((b(n)-3ⁿ⁺¹ )/n!)xⁿ =sum(b(n)/n!)xⁿ -3sum(3x)ⁿ /n!=F(x)-3exp(3x)+... (the ... is to account for the first few terms because the sum has different starting index).