A163986 Sum of all repeated parts of all partitions of n.

0, 0, 2, 4, 13, 26, 55, 92, 161, 253, 401, 595, 901, 1288, 1863, 2611, 3665, 5016, 6895, 9273, 12501, 16591, 22001, 28820, 37753, 48901, 63285, 81217, 104049, 132328, 168061, 212041, 267105, 334654, 418473, 520836, 647101, 800496, 988495, 1216138, 1493441, 1827822, 2233225, 2720138, 3307613, 4010941, 4855577, 5863345, 7069009, 8502628, 10211201

Offset: 0

Author

Omar E. Pol, Aug 14 2009

Keywords

• easy
• nonn

Comments

See A163985 for more information.

Examples

			For n=4, the five partitions of 4 are {(4);(2,2);(3,1);(2,1,1);(1,1,1,1)}. Since 1 and 2 are repeated parts and 3 and 4 are not repeated parts (or isolated parts) then a(4)={(2+2)+(1)+(2+1+1)+(1+1+1+1)}=13.
		

Links

• Omar E. Pol, Illustration of the shell model of partitions (2D view)
• Omar E. Pol, Illustration of the shell model of partitions (3D view)

Crossrefs

Cf. A000041, A066186, A163985.

Programs

Mathematica

Table[Total[Flatten[Select[Split[Sort[Flatten[IntegerPartitions[n]]]], Length[ #]>1&]]],{n,0,50}] (* Harvey P. Dale, Apr 30 2018 *)

Formula

a(0)=0, a(n)=A066186(n)-A163985(n), for n>0.

Extensions

More terms from Alois P. Heinz, Jan 30 2011