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

0,3

Comments

a(n)=Sum(k*A116683(n,k), k>=0).

Links

Table of n, a(n) for n=0..46.

Formula

G.f.=2*product(1+x^j,j=1..infinity)*sum((jx^(2j)/(1+x^(2j)), j=1..infinity)).

Example

a(9)=34 because in the partitions of 9 into distinct parts, namely, [9],[81],[72],[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.

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

Map[Total[Select[Flatten[d[#]], EvenQ]] &, -1 + Range[30]] )
(* Peter J. C. Moses, Mar 14 2014 *)

Crossrefs

Cf. A116681, A116682, A116683.

Sequence in context: A074028 A181926 A061894 * A276057 A116637 A153961

Adjacent sequences: A116681 A116682 A116683 * A116685 A116686 A116687

Keyword

nonn

Author

Emeric Deutsch, Feb 22 2006

Status

approved