Re: [math-fun] help needed with D.E.

I wrote:

I'm not sure that it is helpful, but the ordinary (not exponential) generating function for the A144416 can be written in the form

B(y) = int(exp(y*(x+x^2/2+x^3/6)-x),x=0..infinity) = 1+3*y+31*y^2+842*y^3+45296*y^4+4061871*y^5+...

That follows from A144385 giving formula A144416(n) = int((x+x^2/2+x^3/6)^n*exp(-x), x=0..infinity)/n! Multiplying that by y^n and adding, we get the formula above. Multiplying that by y^n/n! and adding, we get the exponential generating function as A(y) = int(BesselI(0,2*sqrt(y*(x+x^2/2+x^3/6)))*exp(-x),x=0..infinity) = = 1+3*y+31*y^2/2!+842*y^3/3!+45296*y^4/4!+4061871*y^5/5!+... Also doesn't look as if it can be easily simplified. Alec Mihailovs