The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 (list; graph; refs; listen; history; text; internal format)
 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 G. A. Edgar, Table of n, a(n) for n = -1..1003 Michael Somos, Introduction to Ramanujan theta functions 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 Adjacent sequences:  A245418 A245419 A245420 * A245422 A245423 A245424 KEYWORD sign AUTHOR Michael Somos, Jul 21 2014 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified April 15 01:24 EDT 2021. Contains 342974 sequences. (Running on oeis4.)