I would like to be able to enter the following left hand side of an identity. I can write the right hand side (i think) but am not sure about the left. The Left hand side is
$$\sum_{i_1+i_2+...+i_n=k}\binom{k}{i_1,i_2,...,i_n}\frac{f(i_1)f(i_2)...f(i_n)}{k!}$$
where is a function that extracts coefficients from a previously defined generating function. How do you implement this summation with a sum of multiple indicies equal to a particular $k$?
My particular $\,f$ are called Hypergeometric Bernoulli Numbers. The code I have to generate the numbers is below:
T[m_, x_] = Sum[x^j/j!, {j, 0, m}];
g[m_, x_] = x^m/(m! (E^x - T[m - 1, x]));
The hypergeometric bernoulli numbers are extracted using the following
b[m_, n_, M_] := b[m, n, M] = Coefficient[n! Normal[Series[g[m, x], {x, 0, M}]], x, n];
Now I want to sum over the $n$ term defined in the bernoulli number, so I basically want the $\,f$ I wrote in the formula to be replaced by b[m,i[j],M]
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