What is the number of ways of distributing '$n$' identical items among '$r$' identical baskets? I have  seen explanations for distribution among '$r$' groups as opposed to '$r$' identical groups(example: if $12$ eggs are to be distributed among $4$ baskets such that the baskets are not numbered). I know that it cannot be done by finding the number of possible whole number solutions for these $4$ baskets$[(12+4-1)C(4-1)]$, which gives the answer when the baskets are numbered, and then dividing by 4! to "adjust" for the "identical" baskets(as there would also be distributions such as $2,2,3,5; 2,3,2,5; 2,2,2,6$). All this leads me to wonder if there is a single working formula(without the need of step-by-step calculation) for the same.
 A: Using the Polya Enumeration Theorem the closed form is given by
$$[z^n] Z(S_r)\left(\frac{1}{1-z}\right)$$
where $Z(S_r)$ is  the cycle index of the symmetric  group. This cycle
index  is   known  in   species  theory   as  the   multiset  operator
$\mathfrak{M}_{=r}.$
Recall the  recurrence by Lovasz for  the cycle index $Z(S_r)$  of the
set operator $\mathfrak{M}_{=r}$ on $r$ slots, which is
$$Z(S_r) = \frac{1}{r} \sum_{l=1}^r a_l Z(S_{r-l})
\quad\text{where}\quad
Z(S_0) = 1.$$
This recurrence lets us calculate the cycle index $Z(S_r)$ very easily.

For example when $r=5$ the cycle index is
$$Z(S_5) = 
{\frac {{a_{{1}}}^{5}}{120}}+1/12\,a_{{2}}{a_{{1}}}^{3}+1/6\,a_{{3}}
{a_{{1}}}^{2}+1/8\,a_{{1}}{a_{{2}}}^{2}
\\+1/4\,a_{{4}}a_{{1}}+1/6\,a_{{2}}a_{{3}}+1/5\,a_{{5}}$$
and the generating function becomes
$${\frac {1}{120\, \left( 1-z \right) ^{5}}}
+1/12\,{\frac {1}{ \left(-{z}^{2}+1 \right)  \left( 1-z \right) ^{3}}}
+1/6\,{\frac {1}{ \left( -{z}^{3}+1 \right)  \left( 1-z \right) ^{2}}}
\\+1/8\,{\frac {1}{ \left( 1-z \right)  \left( -{z}^{2}+1 \right) ^{2}}}
+1/4\,{\frac {1}{ \left( -{z}^{4}+1 \right)  \left( 1-z \right) }}
+1/6\,{\frac {1}{ \left( -{z}^{2}+1 \right)  \left( -{z}^{3}+1 \right) }}
\\+1/5\, \left( -{z}^{5}+1 \right) ^{-1}.$$
which gives the sequence
$$1, 1, 2, 3, 5, 7, 10, 13, 18, 23, 30, 37, 47, 57, 70, 84, 101, 
\\ 119, 141, 164, 192, \ldots$$
that points to OEIS A001401 which looks to
be a match. Here we have assumed that empty bins are admissible.

This   MSE   link
shows how  one might go about  producing a closed form  expression for
the coefficients of this type of generating function. Furthermore this
MSE       link II
presents an algorithm for computing these closed forms.
There is some Maple code to help explore these data.

pet_varinto_cind :=
proc(poly, ind)
local subs1, subs2, polyvars, indvars, v, pot, res;

    res := ind;

    polyvars := indets(poly);
    indvars := indets(ind);

    for v in indvars do
        pot := op(1, v);

        subs1 :=
        [seq(polyvars[k]=polyvars[k]^pot,
             k=1..nops(polyvars))];

        subs2 := [v=subs(subs1, poly)];

        res := subs(subs2, res);
    od;

    res;
end;

pet_cycleind_symm :=
proc(n)
option remember;

    if n=0 then return 1; fi;

    expand(1/n*add(a[l]*pet_cycleind_symm(n-l), l=1..n));
end;

