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A132180 Expansion of f(q, q^2) * f(-q^3) / f(-q^2)^2 in powers of q where f(, ), f() are Ramanujan theta functions. 4
1, 1, 3, 1, 6, 3, 12, 5, 21, 10, 36, 15, 60, 26, 96, 39, 150, 63, 228, 92, 342, 140, 504, 201, 732, 295, 1050, 415, 1488, 591, 2088, 818, 2901, 1140, 3996, 1554, 5460, 2126, 7404, 2861, 9972, 3855, 13344, 5126, 17748, 6816, 23472, 8970, 30876, 11793, 40413 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
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 eta(q^3)^3 / (eta(q) * eta(q^2) * eta(q^6)) in powers of q.
Euler transform of period 6 sequence [ 1, 2, -2, 2, 1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (v^2 - 2*u)^3 - u^4 * (2*u - 3*v^2) * (4*u - 3*v^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (2/3) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132179.
G.f.: Product_{k>0} (1 + x^k + x^(2*k))^2 / ( (1 + x^k)^2 * (1 - x^k + x^(2*k))).
a(2*n) = A128128(n). a(2*n + 1) = A132302(n).
EXAMPLE
G.f. = 1 + q + 3*q^2 + q^3 + 6*q^4 + 3*q^5 + 12*q^6 + 5*q^7 + 21*q^8 + 10*q^9 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ q^3]^3 / (QPochhammer[ q] QPochhammer[ q^2] QPochhammer[ q^6]), {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -q, q^3] QPochhammer[ -q^2, q^3] QPochhammer[ q^3]^2 / QPochhammer[ q^2]^2, {q, 0, n}]; (* Michael Somos, Nov 01 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^3 / (eta(x + A) * eta(x^2 + A) * eta(x^6 + A)), n))};
CROSSREFS
Sequence in context: A008795 A165188 A294778 * A207630 A368376 A126191
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 12 2007
STATUS
approved

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Last modified March 28 16:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)