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%I #12 Apr 20 2021 19:25:50
%S 1,0,1,2,4,9,20,44,100,225,507,1145,2592,5858,13275,30043,68054,
%T 154132,349182,790954,1792001,4059646,9197535,20837459,47209682,
%U 106957699,242325918,549015961,1243864083,2818122854,6384811753,14465578718,32773596120,74252685312
%N a(n) is the sum over all proper integer partitions with distinct parts of n of the previous terms.
%C By "proper integer partition", one means that the case {n} is excluded for having only one part, equal to the number partitioned.
%C The growth of this function is exponential a(n) -> c * exp(n).
%H Vincenzo Librandi, <a href="/A214952/b214952.txt">Table of n, a(n) for n = 1..70</a>
%F 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.
%e a(6) = (a(5)+a(1)) + (a(4)+a(2)) + (a(3)+a(2)+a(1)) = (4+1) + (2+0) + (1+0+1) = 9.
%t 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}]
%Y Cf. A000041, A214948, A000009.
%K nonn
%O 1,4
%A _Olivier Gérard_, Jul 30 2012
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