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A245421
Expansion of q^(-1) * f(-q^2, -q^7) * f(-q^4, -q^5) / f(-q, -q^8)^2 in powers of q where f() is Ramanujan's two-variable theta function.
2
1, 2, 2, 2, 1, -1, -2, -3, -2, 1, 4, 6, 5, 1, -5, -11, -12, -7, 3, 15, 22, 19, 5, -15, -32, -36, -22, 8, 40, 58, 50, 12, -41, -84, -93, -54, 22, 103, 148, 124, 32, -96, -200, -219, -128, 46, 231, 330, 275, 67, -216, -439, -477, -275, 107, 501, 708, 590, 146
OFFSET
-1,2
COMMENTS
Number 11 of the 15 generalized eta-quotients listed in Table I of Yang 2004.
A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_1(9). [Yang 2004]
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
Y. Yang, Transformation formulas for generalized Dedekind eta functions, Bull. London Math. Soc. 36 (2004), no. 5, 671-682. See p. 679, Table 1.
FORMULA
Expansion of q^(-1) * f(-q) * f(-q^9)^3 / (f(-q^3) * f(-q, -q^8)^3) in powers of q where f() is a Ramanujan theta function.
Euler transform of period 9 sequence [ 2, -1, 0, -1, -1, 0, -1, 2, 0, ...].
Given g.f. A(q), then 0 = f(A(q), A(q^2)) where f(u, v) = (u^2 + v) * (u*v^2 + u - 1) - 2*u*v * (u + v - 1)^2.
Given g.f. A(q), then 0 = f(A(q), A(q^3)) where f(u, v) = (v^2 - v + 1) * (u^3 - v) - 3*u*v * (u - 1) * (2*v - 1);
a(n) = A245424(n) unless n=0.
G.f. T(q) = 1/q + 2 + 2*q + ... for this function is cubically related to T9B(q) of A058091: T9B = T - 2 - 1/T - 1/(T-1). - G. A. Edgar, Apr 13 2017
EXAMPLE
G.f. = 1/q + 2 + 2*q + 2*q^2 + q^3 - q^4 - 2*q^5 - 3*q^6 - 2*q^7 + q^8 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 1/q QPochhammer[ q] / (QPochhammer[ q^3] (QPochhammer[ q^1, q^9] QPochhammer[ q^8, q^9])^3), {q, 0, n}];
a[ n_] := If[ n < -1, 0, With[{m = n + 1}, SeriesCoefficient[ 1/q Product[ (1 - q^k)^{-2, 1, 0, 1, 1, 0, 1, -2, 0}[[Mod[k, 9, 1]]], {k, m}], {q, 0, n}]]];
a[ n_] := SeriesCoefficient[ 1/q QPochhammer[ q^2, q^9] QPochhammer[ q^4, q^9] QPochhammer[ q^5, q^9] QPochhammer[ q^7, q^9] / (QPochhammer[ q^1, q^9] QPochhammer[ q^8, q^9])^2, {q, 0, n}];
PROG
(PARI) {a(n) = if( n<-1, 0, n++; polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[0, -2, 1, 0, 1, 1, 0, 1, -2][k%9 + 1]), n))};
CROSSREFS
Cf. A245424.
Sequence in context: A237593 A338169 A243847 * A134143 A295555 A085684
KEYWORD
sign
AUTHOR
Michael Somos, Jul 21 2014
STATUS
approved