A116684 Sum of the even parts in all partitions of n into distinct parts.

0, 0, 2, 2, 4, 6, 14, 18, 22, 34, 50, 66, 88, 118, 154, 202, 248, 320, 412, 512, 636, 794, 972, 1194, 1454, 1766, 2134, 2576, 3092, 3696, 4426, 5254, 6214, 7364, 8672, 10196, 11986, 14014, 16360, 19084, 22190, 25746, 29860, 34516, 39846, 45952, 52848

Offset: 0

Author

Emeric Deutsch, Feb 22 2006

Keywords

• nonn

Examples

			a(9)=34 because in the partitions of 9 into distinct parts, namely, [9],[8,1],[7,2],[6,3],[6,2,1],[5,4],[5,3,1] and [4,3,2], the sum of the even parts is 8+2+6+6+2+4+4+2=34.
		

Links

• David A. Corneth, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000009, A000700, A066189, A116681, A116682, A116683.

Programs

Maple

f:=2*product(1+x^j,j=1..60)*sum((j*x^(2*j)/(1+x^(2*j)),j=1..35)): fser:=series(f,x=0,55): seq(coeff(fser,x,n),n=0..50);

Mathematica

d[n_] := d[n] = Select[IntegerPartitions[n], DeleteDuplicates[#] == # &]
Map[Total[Select[Flatten[d[#]], EvenQ]] &, -1 + Range[30]]  (* Peter J. C. Moses, Mar 14 2014 *)
(* or *)
CoefficientList[Series[QPochhammer[-1, x]*(EllipticTheta[3, 0, x]^4 + EllipticTheta[4, 0, x]^4 - 2)/48, {x, 0, 100}], x] (* Vaclav Kotesovec, Jun 24 2025 *)

Formula

a(n) = Sum_{k>=0} k*A116683(n,k).

G.f.: 2*(Product_{j>=1} 1+x^j)*(Sum_{j>=1} j*x^(2*j)/(1+x^(2*j))).

A116682(n) + a(n) = A066189(n) = n*A000009(n). - Vaclav Kotesovec, Jun 24 2025

a(n) = Sum_{k=0..floor(n/2)} A000700(n-2*k) * A000009(2*k) * (2*k). - David A. Corneth, Jun 24 2025