A083650 Expansion of f(-x, x^3) * phi(x^2) in powers of x where phi(), f() are Ramanujan theta functions.
1, -1, 2, -1, 0, 2, -1, 0, 0, -2, -1, 2, -2, 0, -2, 1, 0, 2, 0, -2, 0, 1, 0, 0, -2, 0, 0, 0, -1, -2, 2, 0, 2, 0, 0, 2, 3, 0, 0, -2, 0, 0, -2, 0, 2, -1, 2, 0, 0, 0, 2, -2, 0, -2, 2, 1, -2, 2, 0, 0, 0, 0, 0, 0, 0, 2, -1, 0, 0, 0, 0, -2, 2, 0, -2, 2, 0, -2, -1, 0, -2, 0, -2, 0, -2, 0, 0, 4, 0, 0, 0, 1, 0, 0, 0, -2, -2, 0, 0, 0, 2, 2, 0, 0, -2
Offset: 0
Keywords
Examples
G.f. = 1 - x + 2*x^2 - x^3 + 2*x^5 - x^6 - 2*x^9 - x^10 + 2*x^11 - 2*x^12 - 2*x^14 + ... G.f. = q - q^9 + 2*q^17 - q^25 + 2*q^41 - q^49 + 2*q^73 - q^81 + 2*q^89 - 2*q^97 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Programs
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Mathematica
QP = QPochhammer; s = QP[q]*QP[q^2] + O[q]^105; Table[(-1)^Quotient[n, 2]*Coefficient[s, q, n], {n, 0, 105}] (* Jean-François Alcover, Nov 27 2015, adapted from PARI *) -
PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); (-1)^(n\2) * polcoeff( eta(x + A) * eta(x^2 + A), n))}; /* Michael Somos, Mar 02 2010 */
Formula
Euler transform of period 16 sequence [ -1, 2, 1, -2, 1, 1, -1, -3, -1, 1, 1, -2, 1, 2, -1, -2, ...].
G.f.: (Sum_{k>=0} (-1)^(k + [k/4]) * x^(k*(k+1)/2)) * (Sum_k x^(2*k^2)).
(-1)^[n/2] * a(n) = A030204(n).
Extensions
Revised by Michael Somos, Mar 02 2010
Comments