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Andrews's shadow difference function D_3(q).
3

%I #14 Dec 18 2023 10:15:54

%S 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 581,738,924,1155,1435,1785,2199,2717,3332,4084,4987,6076,7375,8949,

%U 10817,13051,15706,18877,22622,27078,32332,38545,45870,54496,64618,76525,90463,106788,125863,148145,174106

%N Andrews's shadow difference function D_3(q).

%C Agrees with A237833 just for n <= 21.

%H 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 Equals (A098151-A275632)/8.

%p F:=(a,q,n)->mul(1-a*q^i,i=0..n-1); # This is (a;q)_n

%p M:=15;

%p # A098151:

%p THETA3:=(add((-1)^n*q^(3*n^2),n=-M..M)) /(add((-1)^n*q^(n^2),n=-M..M));

%p s1:=series(THETA3,q,80); seriestolist(%);

%p # A275632:

%p THETABAR3:=1+2*add( (F(q,q,n-1)*q^(n^2)) / (F(q^n,q,n)*(1-q^n)), n=1..M);

%p s2:=series(THETABAR3,q,80); seriestolist(%);

%p # A275633:

%p series((s1-s2)/8,q,80); seriestolist(%);

%Y Cf. A098151, A237833, A275632.

%K nonn

%O 0,8

%A _N. J. A. Sloane_, Aug 09 2016