Help to Write a Function for Multiple Derivatives

I am new to Mathematica and still struggling to learn basic things. In this spirit, I am trying to compute the following expression:

$$C_n = \left[ \frac{\partial^n W(J)}{\partial J^n}\right]_{J=0},$$ where $$W(J) = \ln Z(J) = \ln \left[\sum_{n=0}^{\infty}\frac{1}{n!} J^n G_n\right]$$, and $$G_n$$s are different non-zero constants.

How can I write a function for computing $$C_n$$ where I would input the value of $$n$$ and get the corresponding expressions? For instance, $$C_1 = G_1$$, $$C_2 = G_2-G_1^2$$, etc. We assume that $$G_0=1$$.

My current approach is very elementary. It is given below.

Z[J_] = g0 + g1 J + g2/2 J^2 + g3/3! J^3 + g4/4! J^4 + g5/5! J^5 + g6/6! J^6
g0 = 1
W[J_] = Log[Z[J]]
C1[J_] = D[W[J], {J, 1}]
C1

Clear["Global`*"]

Z[J_] = g0 + g1 J + g2/2 J^2 + g3/3! J^3 + g4/4! J^4 + g5/5! J^5 + g6/6! J^6;

W[J_] = Log[Z[J]];

The nth derivative of W evaluated at zero is just

c[n_Integer?NonNegative] := c[n] = Derivative[n][W]

The first several are

seq1 = c /@ Range[0, 5] // Simplify

(* {Log[g0], g1/g0, (-g1^2 + g0 g2)/g0^2, (
2 g1^3 - 3 g0 g1 g2 + g0^2 g3)/g0^3, (-6 g1^4 + 12 g0 g1^2 g2 -
4 g0^2 g1 g3 +
g0^2 (-3 g2^2 + g0 g4))/g0^4, (1/(g0^5))(24 g1^5 - 60 g0 g1^3 g2 +
20 g0^2 g1^2 g3 - 5 g0^2 g1 (-6 g2^2 + g0 g4) + g0^3 (-10 g2 g3 + g0 g5))} *)

Alteratively,

c2[n_Integer?NonNegative] := c2[n] = n!*SeriesCoefficient[W[J], {J, 0, n}]

seq2 = c2 /@ Range[0, 5] // Simplify

(* {Log[g0], g1/g0, (-g1^2 + g0 g2)/g0^2, (
2 g1^3 - 3 g0 g1 g2 + g0^2 g3)/g0^3, (-6 g1^4 + 12 g0 g1^2 g2 -
4 g0^2 g1 g3 +
g0^2 (-3 g2^2 + g0 g4))/g0^4, (1/(g0^5))(24 g1^5 - 60 g0 g1^3 g2 +
20 g0^2 g1^2 g3 - 5 g0^2 g1 (-6 g2^2 + g0 g4) + g0^3 (-10 g2 g3 + g0 g5))} *)

seq1 === seq2

(* True *)
• What does this part c2[n_Integer?NonNegative] mean? Aug 30 '20 at 5:49
• It restricts the function to only evaluating for arguments that are nonnegative integers. _Integer requires the head of the argument to be Integer (see documentation for Blank) and ?Nonnegative is a pattern test (see documentation for PatternTest). Aug 30 '20 at 5:56
• Thanks, Bob! I appreciate your help. Aug 30 '20 at 6:25