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 A214341 Expansion of 1 / k(q) = 1 / (r(q) * r(q^2)^2) in powers of q where r() is the Rogers-Ramanujan continued fraction. 2
 1, 1, 2, 1, 1, 0, -1, -2, -2, -1, 1, 3, 4, 4, 1, -2, -6, -8, -7, -3, 4, 10, 14, 12, 6, -6, -16, -22, -20, -8, 8, 26, 34, 31, 12, -14, -41, -54, -47, -20, 23, 61, 84, 72, 31, -32, -90, -122, -107, -44, 45, 133, 174, 154, 61, -68, -192, -254, -220, -90, 100 (list; graph; refs; listen; history; text; internal format)
 OFFSET -1,3 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). Number 12 of the 15 generalized eta-quotients listed in Table I of Yang 2004. - Michael Somos, Aug 07 2014 A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_1(10). [Yang 2004] - Michael Somos, Aug 07 2014 LINKS Seiichi Manyama, Table of n, a(n) for n = -1..10000 S. Cooper, On Ramanujan's function k(q)=r(q)r^2(q^2), Ramanujan J., 20 (2009), 311-328. 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 (1/x) * (f(-x^4, -x^6) * f(-x^3, -x^7)) / (f(-x^2, -x^8) * f(-x, -x^9)) in powers of x where f(,) is Ramanujan's two-variable theta function. Euler transform of period 10 sequence [ 1, 1, -1, -1, 0, -1, -1, 1, 1, 0, ...]. G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u + v)^2 - v * (u^2 - 1). G.f.: (1/x) * Product_{k>0} (1 - x^(10*k - 3)) * (1 - x^(10*k - 4)) * (1 - x^(10*k - 6)) * (1 - x^(10*k - 7)) /((1 - x^(10*k - 1)) * (1 - x^(10*k - 2)) * (1 - x^(10*k - 8)) * (1 - x^(10*k - 9))). Convolution inverse of A112274. a(n) = A112274(n) + A132980(n). EXAMPLE G.f. = 1/q + 1 + 2*q + q^2 + q^3 - q^5 - 2*q^6 - 2*q^7 - q^8 + q^9 + 3*q^10 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ 1/q Product[(1 - q^k)^{-1, -1, 1, 1, 0, 1, 1, -1, -1, 0}[[Mod[k, 10, 1]]], {k, n + 1}], {q, 0, n}]; (* Michael Somos, Aug 07 2014 *) PROG (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( prod( k=1, n, (1 - x^k + A)^[0, -1, -1, 1, 1, 0, 1, 1, -1, -1][k%10 + 1]), n))}; CROSSREFS Cf. A112274, A132980. Sequence in context: A277264 A259538 A099314 * A281871 A131334 A004602 Adjacent sequences:  A214338 A214339 A214340 * A214342 A214343 A214344 KEYWORD sign AUTHOR Michael Somos, Jul 12 2012 STATUS approved

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Last modified October 18 14:52 EDT 2019. Contains 328161 sequences. (Running on oeis4.)