A182724 Sum of all parts of all partitions of n minus the number of partitions of n.

0, 2, 6, 15, 28, 55, 90, 154, 240, 378, 560, 847, 1212, 1755, 2464, 3465, 4752, 6545, 8820, 11913, 15840, 21042, 27610, 36225, 46992, 60900, 78260, 100386, 127820, 162516, 205260, 258819, 324576, 406230, 506022, 629195, 778932, 962555, 1185030

Offset: 1

Author

Omar E. Pol, Jan 30 2011

Keywords

• nonn
• easy

Comments

a(n) is the sum of (the zeroth moments of) all partitions of n minus the partition number of n.

Examples

			a(7) = 90 = (7-1)*15 = 105 - 15, because the number of partitions of 7 is 15 and the sum of all parts of all partitions of 7 is 7*15 = 105.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000041, A066186. Column 1 of A182729.

Programs

Maple

a:= n-> (n-1) *combinat[numbpart](n):
seq (a(n), n =1..50);

Mathematica

pnxt[n_]:=Module[{ps=IntegerPartitions[n]},Total[Flatten[ps]]- Length[ps]]; Array[pnxt,40] (* Harvey P. Dale, Jul 15 2011 *)
Table[(n-1)PartitionsP[n],{n,40}] (* Harvey P. Dale, Jan 17 2015 *)

Formula

a(n) = (n-1)*A000041(n) = A066186(n) - A000041(n).