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A128512
Expansion of q^(-1) * (chi(-q) * chi(-q^9) / chi(-q^3)^2)^6 in powers of q where chi() is a Ramanujan theta function.
1
1, -6, 15, -14, -21, 78, -62, -132, 399, -322, -426, 1332, -964, -1524, 4278, -3072, -4059, 11454, -7802, -11148, 29892, -20284, -26268, 70488, -46341, -62484, 162537, -106340, -135291, 351120, -224958, -292536, 743862, -474370, -594180, 1506060, -946310
OFFSET
-1,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of ((c(q^2)^2 * b(q^3) * c(q^3)) / (c(q)^2 * b(q^6) * c(q^6)))^3 in powers of q where b(), c() are cubic AGM functions.
Expansion of ((eta(q) * eta(q^9) * eta(q^6)^2) / (eta(q^2) * eta(q^18) * eta(q^3)^2))^6 in powers of q.
Euler transform of period 18 sequence [ -6, 0, 6, 0, -6, 0, -6, 0, 0, 0, -6, 0, -6, 0, 6, 0, -6, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u^2 - v) * (w^2 - v) + u*w * (12*(1 + v^2) + 172*v).
G.f.: (1/x) * (Product_{k>0} (1 - x^k + x^(2*k)) / (1 -x^(3*k) + x(6*k)))^6.
EXAMPLE
G.f. = 1/q - 6 + 15*q - 14*q^2 - 21*q^3 + 78*q^4 - 62*q^5 - 132*q^6 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[1/q (QPochhammer[ q] QPochhammer[ q^6]^2 QPochhammer[ q^9] / (QPochhammer[ q^2] QPochhammer[ q^3]^2 QPochhammer[ q^18]))^6, {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x*O(x^n); polcoeff( ( eta(x + A) * eta(x^9 + A) * eta(x^6 + A)^2 / (eta(x^2 + A) * eta(x^18 + A) * eta(x^3 + A)^2))^6, n))};
CROSSREFS
Sequence in context: A070555 A265388 A334352 * A352098 A201142 A200902
KEYWORD
sign
AUTHOR
Michael Somos, Mar 05 2007
STATUS
approved