cp's OEIS Frontend

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

%I A116684 #20 Jun 25 2025 01:09:31
%S A116684 0,0,2,2,4,6,14,18,22,34,50,66,88,118,154,202,248,320,412,512,636,794,
%T A116684 972,1194,1454,1766,2134,2576,3092,3696,4426,5254,6214,7364,8672,
%U A116684 10196,11986,14014,16360,19084,22190,25746,29860,34516,39846,45952,52848
%N A116684 Sum of the even parts in all partitions of n into distinct parts.
%H A116684 David A. Corneth, <a href="/A116684/b116684.txt">Table of n, a(n) for n = 0..10000</a>
%F A116684 a(n) = Sum_{k>=0} k*A116683(n,k).
%F A116684 G.f.: 2*(Product_{j>=1} 1+x^j)*(Sum_{j>=1} j*x^(2*j)/(1+x^(2*j))).
%F A116684 A116682(n) + a(n) = A066189(n) = n*A000009(n). - _Vaclav Kotesovec_, Jun 24 2025
%F A116684 a(n) = Sum_{k=0..floor(n/2)} A000700(n-2*k) * A000009(2*k) * (2*k). - _David A. Corneth_, Jun 24 2025
%e A116684 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.
%p A116684 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);
%t A116684 d[n_] := d[n] = Select[IntegerPartitions[n], DeleteDuplicates[#] == # &]
%t A116684 Map[Total[Select[Flatten[d[#]], EvenQ]] &, -1 + Range[30]]  (* _Peter J. C. Moses_, Mar 14 2014 *)
%t A116684 (* or *)
%t A116684 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 *)
%Y A116684 Cf. A000009, A000700, A066189, A116681, A116682, A116683.
%K A116684 nonn
%O A116684 0,3
%A A116684 _Emeric Deutsch_, Feb 22 2006