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A260118
Expansion of f(-x, -x^5) * psi(x^4) in powers of x where psi(), f() are Ramanujan theta functions.
4
1, -1, 0, 0, 1, -2, 0, 0, 1, -1, 0, 0, 2, -1, 0, 0, 1, -1, 0, 0, 2, -1, 0, 0, 1, -2, 0, 0, 1, -1, 0, 0, 1, -2, 0, 0, 0, -1, 0, 0, 3, -1, 0, 0, 1, -3, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, -1, 0, 0, 2, -2, 0, 0, 1, -2, 0, 0, 2, -1, 0, 0, 0, -1, 0, 0, 1, -1, 0, 0, 2
OFFSET
0,6
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of psi(-x^2)^2 * psi(x^3) / f(x) in powers of x where psi(), f() are Ramanujan theta functions.
Expansion of psi(-x) * psi(x^3) / chi(-x^4)^2 in powers of x where psi(), chi() are Ramanujan theta functions.
Expansion of q^(-5/6) * eta(q) * eta(q^6)^2 * eta(q^8)^2 / (eta(q^2) * eta(q^3) * eta(q^4)) in powers of q.
Euler transform of period 24 sequence [ -1, 0, 0, 1, -1, -1, -1, -1, 0, 0, -1, 0, -1, 0, 0, -1, -1, -1, -1, 1, 0, 0, -1, -2, ...].
a(n) = (-1)^n * A128582(n). 2 * a(n) = - A134177(3*n + 2) = A190615(3*n + 2).
a(4*n) = A128583(n). a(4*n + 1) = - A128591(n). a(4*n + 2) = a(4*n + 3) = 0.
EXAMPLE
G.f. = 1 - x + x^4 - 2*x^5 + x^8 - x^9 + 2*x^12 - x^13 + x^16 - x^17 + ...
G.f. = q^5 - q^11 + q^29 - 2*q^35 + q^53 - q^59 + 2*q^77 - q^83 + q^101 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, x]^2 EllipticTheta[ 2, 0, x^(3/2)]/(4 x^(7/8) QPochhammer[ -x]), {x, 0, n}];
a[ n_] := If[ n < 0, 0, With[ {m = 6 n + 5}, (-1)^n DivisorSum[ m, {-1, 0, 0, 0, 1, 0}[[Mod[#, 6, 1]]] {1, 0, 0, 0, 0, 0, 1, 0}[[Mod[m/#, 8, 1]]] &]]];
PROG
(PARI) {a(n) = my(m); if( n<0, 0, m=6*n + 5; (-1)^n * sumdiv(m, d, ((d%6==5) - (d%6==1)) * ((m/d%8==1) + (m/d%8==7))))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^2 * eta(x^8 + A)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^4 + A)), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Jul 16 2015
STATUS
approved