A072834 Exponents occurring in expansion of F_8(q^2).

0, 7, 12, 15, 16, 23, 28, 31, 32, 39, 44, 47, 48, 55, 60, 63, 64, 71, 76, 79, 80, 87, 92, 95, 96, 103, 108, 111, 112, 119, 124, 127, 128, 135, 140, 143, 144, 151, 156, 159, 160, 167, 172, 175, 176, 183, 188, 191, 192, 199, 204, 207, 208, 215, 220, 223, 224, 231, 236, 239, 240, 247

Offset: 0

Author

N. J. A. Sloane, Jul 25 2002

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• S. Ahlgren, The sixth, eighth, ninth and tenth powers of Ramanujan's theta function, Proc. Amer. Math. Soc. 128 (2000), 1333-1338.

Crossrefs

Cf. A023919.

Programs

Mathematica

f[x_, y_]:= QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; F[8, q_]:= ( f[q^4, q^4]^7 + (2*q*f[q^8, q^24])^7 + 14*f[q^2, q^2]^2*f[q^4, q^4]^5 - 7*f[q^4, q^4]^3*f[q^2, q^2]^4)/8; cfs = CoefficientList[Series[F[6, q], {q, 0, 500}], q]; Take[Pick[Range[Length[cfs]] - 1, Sign[Abs[cfs]], 1], 50] (* G. C. Greubel, Apr 16 2018 *)

Formula

Empirical g.f.: x*(x^3+3*x^2+5*x+7) / (x^5-x^4-x+1). - Colin Barker, Jul 31 2013

Conjecture: a(n) = 4*n + pa(n mod 4), where pa(k) = 0,3,4,3 for k=0,1,2,3 respectively; lim_{n->infinity} a(n)/n = 4; a(n)/n >= 4; a(n+4) = a(n) + 16. - Jerzy R Borysowicz, Jan 16 2022

Extensions

Terms a(25) onward added by G. C. Greubel, Apr 16 2018