OFFSET
1,4
COMMENTS
By "proper integer partition", one means that the case {n} is excluded for having only one part, equal to the number partitioned.
The growth of this function is exponential a(n) -> c * exp(n).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..70
FORMULA
a(n) = sum( sum( a(i), i in p) , p in P*(n)) where Q*(n) is the set of all integer partitions of n with distinct parts excluding {n}, p is a partition of Q*(n), i is a part of p.
EXAMPLE
a(6) = (a(5)+a(1)) + (a(4)+a(2)) + (a(3)+a(2)+a(1)) = (4+1) + (2+0) + (1+0+1) = 9.
MATHEMATICA
Clear[a]; a[1] := 1; a[n_Integer] := a[n] = Plus @@ Map[Function[p, Plus @@ Map[a, p]], Select[Drop[IntegerPartitions[n], 1], Union[#]==Sort[#]&]]; Table[ a[n], {n, 1, 30}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Olivier Gérard, Jul 30 2012
STATUS
approved