A116682 Sum of the odd parts in all partitions of n into distinct parts.

0, 1, 0, 4, 4, 9, 10, 17, 26, 38, 50, 66, 92, 116, 154, 203, 264, 326, 416, 514, 644, 802, 986, 1198, 1474, 1784, 2156, 2608, 3124, 3728, 4454, 5286, 6266, 7420, 8736, 10279, 12062, 14106, 16472, 19214, 22330, 25914, 30032, 34714, 40058, 46208, 53136

Offset: 0

Author

Emeric Deutsch, Feb 22 2006

Keywords

• nonn

Examples

			a(9)=38 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 odd parts is 9+1+7+3+1+5+5+3+1+3=38.
		

Links

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

Crossrefs

Cf. A000009, A000700, A066189, A116681, A116683, A116684.

Programs

Maple

f:=product(1+x^j,j=1..70)*sum((2*j-1)*x^(2*j-1)/(1+x^(2*j-1)),j=1..40): fser:=series(f,x=0,60): seq(coeff(fser,x,n),n=0..50);

Mathematica

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

Formula

a(n) = Sum_{k=0..n} k*A116681(n,k).

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

a(n) + A116684(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) * (n - 2*k). - David A. Corneth, Jun 24 2025