OFFSET
0,2
COMMENTS
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2500 from G. C. Greubel)
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of (eta(q^2) * eta(q^16))^5 / (eta(q) * eta(q^4) * eta(q^8) * eta(q^32))^2 in powers of q.
Euler transform of period 32 sequence [2, -3, 2, -1, 2, -3, 2, 1, 2, -3, 2, -1, 2, -3, 2, -4, 2, -3, 2, -1, 2, -3, 2, 1, 2, -3, 2, -1, 2, -3, 2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 32^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
G.f.: (Sum_{k in Z} x^k^2) * (Sum_{k in Z} (x^8)^k^2).
a(4*n + 2) = a(4*n + 3) = a(8*n + 5) = 0. a(4*n) = a(8*n) = A033715(n). a(4*n + 1) = A033715(4*n + 1). a(8*n + 1) = 2 * A112603(n). a(8*n + 4) = 2 * A113411(n).
(-1)^n * a(n) = A242609(n). - Michael Somos, Feb 20 2015
EXAMPLE
G.f. = 1 + 2*q + 2*q^4 + 2*q^8 + 6*q^9 + 4*q^12 + 2*q^16 + 4*q^17 + 4*q^24 + 2*q^25 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^8], {q, 0, n}];
PROG
(PARI) {a(n) = if( n<1, n==0, 2 * (n%4 < 2) * sumdiv( n, d, kronecker( -2, d)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^16 + A))^5 / (eta(x + A) * eta(x^4 + A) * eta(x^8 + A) * eta(x^32 + A))^2, n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, May 31 2013
STATUS
approved