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

0,4

Links

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

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

Example

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.

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 *)

Crossrefs

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

Sequence in context: A332777 A238629 A192032 * A168157 A222045 A394184

Adjacent sequences: A116679 A116680 A116681 * A116683 A116684 A116685

Keyword

nonn

Author

Emeric Deutsch, Feb 22 2006

Status

approved