# Using Derivative with list in curly brackets

I'd like to make a formula for a sum containing Derivative where the number of arguments in the latter depends on an input parameter n. Also involved is a list c calculated before and a function f that has n inputs. Here's the version for n=2

c[[1,2]]*Derivative[c[[1,1,1],c[[1,1,2]]]][f][z1,z2]+
c[[2,2]]*Derivative[c[[2,1,1],c[[2,1,2]]]][f][z1,z2]+
c[[3,2]]*Derivative[c[[3,1,1],c[[3,1,2]]]][f][z1,z2];


And here for n=3:

c[[1,2]]*Derivative[c[[1,1,1],c[[1,1,2],c[[1,1,3]]]][f][z1,z2,z3]+
c[[2,2]]*Derivative[c[[2,1,1],c[[2,1,2],c[[2,1,3]]]][f][z1,z2,z3]+
c[[3,2]]*Derivative[c[[3,1,1],c[[3,1,2],c[[3,1,3]]]][f][z1,z2,z3]+
c[[4,2]]*Derivative[c[[4,1,1],c[[4,1,2],c[[4,1,3]]]][f][z1,z2,z3]+
c[[5,2]]*Derivative[c[[5,1,1],c[[1,1,2],c[[5,1,3]]]][f][z1,z2,z3]+
c[[6,2]]*Derivative[c[[6,1,1],c[[6,1,2],c[[6,1,3]]]][f][z1,z2,z3];


And so on. I was thinking about how I can make this more general and I think I can almost get it working using sum, doing something like (for n=4)

Sum[c[[i,2]]]*Derivative[Table[c[[i,1,j]],{j,1,4}]][f][z1,z2,z3,z4],{i,1,24}];


The problem with this is that Table outputs a list with curly brackets, and this won't work with Derivative, as seen from e.g. running

f[x_,y_]=x^2+5*x*y;
Derivative[1,1][f][x,y]
Derivative[{1,1}][f][x,y]


The first way works but not the second. Trying to get rid of the brackets I've tried Flatten with different inputs but that seems to only remove brackets inside brackets; I can't get rid of the outermost ones.

Does anyone know a way to get rid of the brackets from a Table, or use Derivative with a list in curly brackets? Or maybe a better way to do the whole thing?

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If you want n! terms, this is the thing:

expr[n_] := Sum[c[[i, 2]] Derivative[Sequence @@ Table[c[[i, 1, j]],
{j, n}]][f][##], {i, n!}] & @@ ToExpression["z" <> ToString[#] & /@ Range[n]]


I'm not sure, but with 24 for n=4, I'm kinda convinced, even though you have 3 terms for n=2.

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That works perfectly, thank you very much! And yeah it is supposed to be n!, I have a typo in the n=2 example in the OP. – jorgen Feb 28 '14 at 17:36