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%I #23 Dec 18 2023 12:11:38
%S 1,2,1,1,2,-1,1,3,-2,1,2,0,2,1,0,-1,5,2,-1,2,-3,5,3,-1,2,0,1,1,2,-2,2,
%T 5,2,-4,0,1,-1,6,0,4,-3,-1,3,-1,2,0,4,-2,2,4,-2,1,5,-2,-2,-2,4,6,1,3,
%U -2,4,-3,-1,-2,4,6,2,0,-4,5,1,3,-1,0,0,4,-1,-2,4,-2,2,5,2,5,-5,-2,6,-4,0,-3
%N Weighted count of partitions of 2n+1 into odd parts in which the largest part appears an odd number of times and all other parts appear twice, with respect to a certain weight.
%C See Andrews (2016) for the definition of the particular weight used here.
%C Andrews (2016), Theorem 2, shows that A008443(n) = A290735(n) + a(n) + A290739(n).
%H George E. Andrews, <a href="https://georgeandrews1.github.io/pdf/320.pdf">The Bhargava-Adiga Summation and Partitions</a>, 2016. See Lemma 3.3.
%F See Maple code for g.f.
%p M:=201;
%p B:=proc(a, q, n) local j, t1; global M; t1:=1;
%p for j from 0 to M do t1:=t1*(1-a*q^j)/(1-a*q^(n+j)); od;
%p t1; end;
%p D2:=add( q^(2*m+1)*B(q^2,q^4,m)/(1-q^(4*m+2)), m=0..M):
%p series(D2,q,M); d2seq:=seriestolist(%); BISECT(%,1);
%Y Cf. A008443, A290733-A290740.
%K sign
%O 0,2
%A _N. J. A. Sloane_, Aug 10 2017
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