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A275632 Andrews's 4-shadow function THETA.BAR_3(q).

%I A275632 #17 Dec 18 2023 10:12:38
%S A275632 1,2,4,6,10,8,16,12,20,18,24,16,38,20,32,32,42,24,52,28,56,44,48,32,
%T A275632 80,42,56,54,76,40,96,44,84,64,72,64,122,52,80,76,120,56,128,60,112,
%U A275632 104,96,64,166,78,124,96,132,72,160,96
%N A275632 Andrews's 4-shadow function THETA.BAR_3(q).
%H A275632 R. J. Mathar, <a href="/A275632/b275632.txt">Table of n, a(n) for n = 0..999</a>
%H A275632  George E. Andrews, <a href="https://georgeandrews1.github.io/pdf/315.pdf">4-Shadows in q-Series and the Kimberling Index</a>, Preprint, May 15, 2016
%F A275632 See Section 1 of Andrews (2016) or the Maple code below.
%p A275632 F:=(a,q,n)->mul(1-a*q^i,i=0..n-1); # This is (a;q)_n
%p A275632 M:=15;
%p A275632 # A098151:
%p A275632 THETA3:=(add((-1)^n*q^(3*n^2),n=-M..M)) /(add((-1)^n*q^(n^2),n=-M..M));
%p A275632 s1:=series(THETA3,q,80); seriestolist(%);
%p A275632 # A275632:
%p A275632 THETABAR3:=1+2*add( (F(q,q,n-1)*q^(n^2)) / (F(q^n,q,n)*(1-q^n)), n=1..M);
%p A275632 s2:=series(THETABAR3,q,80); seriestolist(%);
%p A275632 # A275633:
%p A275632 series((s1-s2)/8,q,80); seriestolist(%);
%Y A275632 Cf. A098151, A275633.
%K A275632 nonn
%O A275632 0,2
%A A275632 _N. J. A. Sloane_, Aug 09 2016