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A275633 Andrews's shadow difference function D_3(q).

%I A275633 #14 Dec 18 2023 10:15:54
%S A275633 0,0,0,0,0,1,1,3,4,7,10,16,20,31,41,56,74,101,129,172,219,284,363,464,
%T A275633 581,738,924,1155,1435,1785,2199,2717,3332,4084,4987,6076,7375,8949,
%U A275633 10817,13051,15706,18877,22622,27078,32332,38545,45870,54496,64618,76525,90463,106788,125863,148145,174106
%N A275633 Andrews's shadow difference function D_3(q).
%C A275633 Agrees with A237833 just for n <= 21.
%H A275633 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 A275633 Equals (A098151-A275632)/8.
%p A275633 F:=(a,q,n)->mul(1-a*q^i,i=0..n-1); # This is (a;q)_n
%p A275633 M:=15;
%p A275633 # A098151:
%p A275633 THETA3:=(add((-1)^n*q^(3*n^2),n=-M..M)) /(add((-1)^n*q^(n^2),n=-M..M));
%p A275633 s1:=series(THETA3,q,80); seriestolist(%);
%p A275633 # A275632:
%p A275633 THETABAR3:=1+2*add( (F(q,q,n-1)*q^(n^2)) / (F(q^n,q,n)*(1-q^n)), n=1..M);
%p A275633 s2:=series(THETABAR3,q,80); seriestolist(%);
%p A275633 # A275633:
%p A275633 series((s1-s2)/8,q,80); seriestolist(%);
%Y A275633 Cf. A098151, A237833, A275632.
%K A275633 nonn
%O A275633 0,8
%A A275633 _N. J. A. Sloane_, Aug 09 2016