Golf the Inverse of Exponential Generating Function

Question

Earlier, we talked about exponential generating functions (e.g.f.).

Task

You will take a few terms of a sequence.

Your task is to find another sequence with that many terms, whose e.g.f., when multiplied to the original e.g.f., would be exactly the constant function 1 accurate to that many terms.

That is, given a0=1, a1, a2, ..., an, find b0=1, b1, b2, ..., bn such that a0 + a1x + a2x^2/2! + a3x^3/3! + ... + anx^n/n! multiplied by b0 + b1x + b2x^2/2! + b3x^3/3! + ... + bnx^n/n! equals 1 + O(x^(n+1)).

Specs

• They will start at the zeroth power and will be consecutive.
• The first term is guaranteed to be 1.
• All the terms are guaranteed to be integers.
• There will not be negative powers.

Testcases

Input : 1,0,-1,0,1,0,-1,0 (this is actually cos x)
Output: 1,0, 1,0,5,0,61,0 (this is actually sec x)

Input : 1, 2, 4, 8, 16, 32 (e^(2x))
Output: 1,-2, 4,-8, 16,-32 (e^(-2x))

Input : 1,-1,0,0, 0,  0,  0 (1-x)
Output: 1, 1,2,6,24,120,720 (1/(1-x) = 1+x+x^2+...)

Answer 1

Julia, 91 81 77 bytes

!v=(b=[1;0v[r=2:end]];for i=r,j=0:i-2 b[i]-=binomial(i-1,j)v[i-j]b[j+1]end;b)

Usage: ![1 0 -1 0 1 0 -1 0]

Saved 4 bytes thanks to Dennis!

This is a fairly straightforward implementation - it uses the fact that the "exponential" part causes a binomial to appear in the solution. Otherwise, it's just solving for the relevant coefficients.