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 A116682 Sum of the odd parts in all partitions of n into distinct parts. 3

%I #6 Mar 14 2014 12:32:18

%S 0,1,0,4,4,9,10,17,26,38,50,66,92,116,154,203,264,326,416,514,644,802,

%T 986,1198,1474,1784,2156,2608,3124,3728,4454,5286,6266,7420,8736,

%U 10279,12062,14106,16472,19214,22330,25914,30032,34714,40058,46208,53136

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

%C a(n)=Sum(k*A116681(n,k), k=0..n).

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

%e 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 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 d[n_] := d[n] = Select[IntegerPartitions[n], DeleteDuplicates[#] == # &]

%t Map[Total[Select[Flatten[d[#]], OddQ]] &, -1 + Range[30]] (* Peter J. C. Moses, Mar 14 2014 *)

%Y Cf. A116681, A116683, A116684.

%K nonn

%O 0,4

%A _Emeric Deutsch_, Feb 22 2006

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Last modified August 10 09:30 EDT 2024. Contains 375044 sequences. (Running on oeis4.)