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 A116684 Sum of the even parts in all partitions of n into distinct parts. 3

%I #6 Mar 14 2014 12:32:26

%S 0,0,2,2,4,6,14,18,22,34,50,66,88,118,154,202,248,320,412,512,636,794,

%T 972,1194,1454,1766,2134,2576,3092,3696,4426,5254,6214,7364,8672,

%U 10196,11986,14014,16360,19084,22190,25746,29860,34516,39846,45952,52848

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

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

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

%e 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.

%p 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 Map[Total[Select[Flatten[d[#]], EvenQ]] &, -1 + Range[30]] )

%t (* Peter J. C. Moses, Mar 14 2014 *)

%Y Cf. A116681, A116682, A116683.

%K nonn

%O 0,3

%A _Emeric Deutsch_, Feb 22 2006

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