OFFSET
0,2
COMMENTS
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
FORMULA
Expansion of (f(-x^6)^2 * chi(x^3) / chi(x))^2 in powers of x where chi(), f() are Ramanujan theta functions.
Expansion of q^(-5/6) * (eta(q) * eta(q^4) * eta(q^6)^4 / (eta(q^2)^2 * eta(q^3) * eta(q^12)))^2 in powers of q.
Euler transform of period 12 sequence [-2, 2, 0, 0, -2, -4, -2, 0, 0, 2, -2, -4, ...].
Convolution square of A258832.
EXAMPLE
G.f. = 1 - 2*x + 3*x^2 - 4*x^3 + 5*x^4 - 8*x^5 + 7*x^6 - 8*x^7 + 9*x^8 + ...
G.f. = q^5 - 2*q^11 + 3*q^17 - 4*q^23 + 5*q^29 - 8*q^35 + 7*q^41 - 8*q^47 + ...
MAPLE
with(numtheory):
seq((-1)^(n-1)*sigma(6*n - 1)/6, n=1..10^3); # Muniru A Asiru, Jan 30 2018
MATHEMATICA
a[ n_] := If[ n < 0, 0, (-1)^n DivisorSigma[ 1, 6 n + 5] / 6];
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^6]^2 QPochhammer[ x, -x] / QPochhammer[ x^3, -x^3])^2, {x, 0, n}];
Table[(-1)^n DivisorSigma[1, 6 n + 5] / 6, {n, 0, 60}] (* Vincenzo Librandi, Jan 30 2018 *)
PROG
(PARI) {a(n) = if(n<0, 0, (-1)^n*sigma(6*n+5)/6)};
(PARI) {a(n) = my(A); if(n<0, 0, A = x*O(x^n); polcoeff((eta(x + A)*eta(x^4 + A)*eta(x^6 + A)^4/(eta(x^2+A)^2*eta(x^3+A)*eta(x^12+A)))^2, n))};
(GAP) List([0..70], n -> (-1)^n*Sigma(6*n+5)/6); # Muniru A Asiru, Jan 30 2018
(Magma) [(-1)^n*SumOfDivisors(6*n+5)/6: n in [0..70]]; // Vincenzo Librandi, Jan 30 2018
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Jun 11 2015
STATUS
approved