A214952 a(n) is the sum over all proper integer partitions with distinct parts of n of the previous terms.

1, 0, 1, 2, 4, 9, 20, 44, 100, 225, 507, 1145, 2592, 5858, 13275, 30043, 68054, 154132, 349182, 790954, 1792001, 4059646, 9197535, 20837459, 47209682, 106957699, 242325918, 549015961, 1243864083, 2818122854, 6384811753, 14465578718, 32773596120, 74252685312

Offset: 1

Author

Olivier Gérard, Jul 30 2012

Keywords

• nonn

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).

Examples

			a(6) = (a(5)+a(1)) + (a(4)+a(2)) + (a(3)+a(2)+a(1)) = (4+1) + (2+0) + (1+0+1) = 9.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..70

Crossrefs

Cf. A000041, A214948, A000009.

Programs

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}]

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.