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A290735 a(n) = weighted sum over all the self-conjugate partitions of 4n + 1 into odd parts, with respect to a certain weight.

%I A290735 #23 Dec 18 2023 12:08:01
%S A290735 1,2,2,3,4,3,5,6,4,6,7,6,7,8,6,7,11,7,8,10,6,11,12,7,10,12,8,11,13,8,
%T A290735 11,16,10,9,15,8,13,18,9,14,14,10,15,16,10,13,20,11,13,20,8,17,22,8,
%U A290735 14,17,15,18,20,12,14,23,12,14,20,12,21,25,9,16,22,14,21,22,12,15,26,16,14,26
%N A290735 a(n) = weighted sum over all the self-conjugate partitions of 4n + 1 into odd parts, with respect to a certain weight.
%C A290735 See Andrews (2016) for the definition of the particular weight used here.
%C A290735 Andrews (2016), Theorem 2, shows that A008443(n) = a(n) + A290737(n) + A290739(n).
%C A290735 Andrews conjectures that a(n) > 0 for all n. The conjecture is known to be true for n <= 1000.
%C A290735 Andrews also conjectures that a(n) > |A290737(n) + A290739(n)| for n >= 2 (see A290740).
%H A290735 George E. Andrews, <a href="https://georgeandrews1.github.io/pdf/320.pdf">The Bhargava-Adiga Summation and Partitions</a>, 2016. See Lemma 3.2.
%F A290735 See Maple code for g.f.
%p A290735 M:=101;
%p A290735 B:=proc(a,q,n) local j,t1; global M; t1:=1;
%p A290735 for j from 0 to M do t1:=t1*(1-a*q^j)/(1-a*q^(n+j)); od;
%p A290735 t1; end;
%p A290735 D1:=add( (-1)^m*q^(m*(m+1))/(B(q,q^2,m+1)*(1-q^(2*m+1))), m=0..M):
%p A290735 series(D1,q,M); d1seq:=seriestolist(%);
%t A290735 M = 101;
%t A290735 B[a_, q_, n_] := Module[{j, t1},  t1 = 1; For[j = 0, j <= M, j++, t1 = t1*(1-a*q^j)/(1-a*q^(n+j))]; t1];
%t A290735 D1 = Sum[(-1)^m*q^(m*(m+1))/(B[q, q^2, m+1]*(1-q^(2*m+1))), {m, 0, M}];
%t A290735 Series[D1, {q, 0, M}] // CoefficientList[#, q]& (* _Jean-François Alcover_, Mar 16 2023, after Maple code *)
%Y A290735 Cf. A008443, A290733, A290734, A290736, A290737, A290738, A290739, A290740.
%K A290735 nonn
%O A290735 0,2
%A A290735 _N. J. A. Sloane_, Aug 10 2017