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Sum of the sizes of the Durfee squares of all partitions of n into odd parts.
1

%I #4 Oct 06 2013 14:02:36

%S 1,1,2,2,3,5,6,8,12,16,20,27,34,43,56,68,84,104,126,153,187,222,266,

%T 317,378,445,528,620,728,853,997,1159,1353,1566,1818,2102,2427,2793,

%U 3218,3692,4236,4849,5545,6325,7220,8210,9337,10599,12023,13609,15403,17394

%N Sum of the sizes of the Durfee squares of all partitions of n into odd parts.

%C a(n)=sum(k*A116464(n,k), k>=1).

%F G.f.=sum(2ix^(4i^2+2i)/[product(1-x^(2j),j=1..2i)product(1-x^(2j-1),j=1..i)], i=1..infinity)+ sum((2i-1)x^((2i-1)^2)/[product(1-x^(2j),j=1..2i-1)product(1-x^(2j-1),j=1..i)],i=1..infinity).

%e a(7)=6 because the partitions of 5 into odd parts are [7], [5,1,1], [3,3,1],

%e [3,1,1,1,1] and [1,1,1,1,1,1,1], having Durfee squares of sizes 1, 1, 2, 1 and 1, respectively.

%p g:=sum(2*i*x^(4*i^2+2*i)/product(1-x^(2*j),j=1..2*i)/product(1-x^(2*j-1),j=1..i),i=1..30)+ sum((2*i-1)*x^((2*i-1)^2)/product(1-x^(2*j),j=1..2*i-1)/product(1-x^(2*j-1),j=1..i),i=1..30): gser:=series(g,x=0,62): seq(coeff(gser,x^n),n=1..60);

%Y Cf. A116464.

%K nonn

%O 1,3

%A _Emeric Deutsch_ and _Vladeta Jovovic_, Feb 18 2006