cp's OEIS Frontend

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

%I A116682 #19 Jun 25 2025 01:09:25
%S A116682 0,1,0,4,4,9,10,17,26,38,50,66,92,116,154,203,264,326,416,514,644,802,
%T A116682 986,1198,1474,1784,2156,2608,3124,3728,4454,5286,6266,7420,8736,
%U A116682 10279,12062,14106,16472,19214,22330,25914,30032,34714,40058,46208,53136
%N A116682 Sum of the odd parts in all partitions of n into distinct parts.
%H A116682 David A. Corneth, <a href="/A116682/b116682.txt">Table of n, a(n) for n = 0..10000</a>
%F A116682 a(n) = Sum_{k=0..n} k*A116681(n,k).
%F A116682 G.f.: (Product_{j>=1} 1+x^j)*(Sum_{j>=1} (2*j-1)*x^(2*j-1)/(1+x^(2*j-1))).
%F A116682 a(n) + A116684(n) = A066189(n) = n*A000009(n). - _Vaclav Kotesovec_, Jun 24 2025
%F A116682 a(n) = Sum_{k=0..floor(n/2)} A000700(n-2*k) * A000009(2*k) * (n - 2*k). - _David A. Corneth_, Jun 24 2025
%e A116682 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.
%p A116682 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);
%t A116682 d[n_] := d[n] = Select[IntegerPartitions[n], DeleteDuplicates[#] == # &]
%t A116682 Map[Total[Select[Flatten[d[#]], OddQ]] &, -1 + Range[30]]  (* _Peter J. C. Moses_, Mar 14 2014 *)
%t A116682 (* or *)
%t A116682 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 *)
%Y A116682 Cf. A000009, A000700, A066189, A116681, A116683, A116684.
%K A116682 nonn
%O A116682 0,4
%A A116682 _Emeric Deutsch_, Feb 22 2006