OFFSET
1,2
COMMENTS
If a collapse is a joining of some number of equal parts of an integer partition p, we say p is collapsible if by some sequence of collapses it can be reduced to a single part. An example of such a sequence of collapses is (32211111)->(332211)->(33222)->(6222)->(66)->(n) which shows that (32211111) is a collapsible partition of n=twelve.
LINKS
FORMULA
a(2^n)=A002577(n+1).
MATHEMATICA
repcaps[q_List]:=repcaps[q]=Union[{q}, If[UnsameQ@@q, {}, Union@@repcaps/@Union[Sort[Append[Drop[q, #], Plus@@Take[q, #]], Greater]&/@Select[Tuples[Range[Length[q]], 2], And[Less@@#, SameQ@@Take[q, #]]&]]]];
repenum[n_]:=Length[Select[IntegerPartitions[n], MemberQ[repcaps[#], {n}]&]];
Table[repenum[n], {n, 1, 32}](* Gus Wiseman, Aug 11 2016 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 11 2016
STATUS
approved