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

0, 0, 0, 0, 0, 1, 1, 3, 4, 7, 10, 16, 20, 31, 41, 56, 74, 101, 129, 172, 219, 284, 363, 464, 581, 738, 924, 1155, 1435, 1785, 2199, 2717, 3332, 4084, 4987, 6076, 7375, 8949, 10817, 13051, 15706, 18877, 22622, 27078, 32332, 38545, 45870, 54496, 64618, 76525, 90463, 106788, 125863, 148145, 174106

Offset

0,8

Comments

Agrees with A237833 just for n <= 21.

Links

Table of n, a(n) for n=0..54.

George E. Andrews, 4-Shadows in q-Series and the Kimberling Index, Preprint, May 15, 2016.

Formula

Equals (A098151-A275632)/8.

Maple

F:=(a, q, n)->mul(1-a*q^i, i=0..n-1); # This is (a; q)_n
M:=15;
# A098151:
THETA3:=(add((-1)^n*q^(3*n^2), n=-M..M)) /(add((-1)^n*q^(n^2), n=-M..M));
s1:=series(THETA3, q, 80); seriestolist(%);
# A275632:
THETABAR3:=1+2*add( (F(q, q, n-1)*q^(n^2)) / (F(q^n, q, n)*(1-q^n)), n=1..M);
s2:=series(THETABAR3, q, 80); seriestolist(%);
# A275633:
series((s1-s2)/8, q, 80); seriestolist(%);

Crossrefs

Cf. A098151, A237833, A275632.

Sequence in context: A147871 A368896 A237833 * A004397 A324368 A241654

Adjacent sequences: A275630 A275631 A275632 * A275634 A275635 A275636

Keyword

nonn

Author

N. J. A. Sloane, Aug 09 2016

Status

approved