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%I #26 Mar 12 2021 22:24:46
%S 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,
%T -22,-20,-8,8,26,34,31,12,-14,-41,-54,-47,-20,23,61,84,72,31,-32,-90,
%U -122,-107,-44,45,133,174,154,61,-68,-192,-254,-220,-90,100
%N Expansion of 1 / k(q) = 1 / (r(q) * r(q^2)^2) in powers of q where r() is the Rogers-Ramanujan continued fraction.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Number 12 of the 15 generalized eta-quotients listed in Table I of Yang 2004. - _Michael Somos_, Aug 07 2014
%C A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_1(10). [Yang 2004] - _Michael Somos_, Aug 07 2014
%H Seiichi Manyama, <a href="/A214341/b214341.txt">Table of n, a(n) for n = -1..10000</a>
%H S. Cooper, <a href="https://dx.doi.org/10.1007/s11139-009-9198-5">On Ramanujan's function k(q)=r(q)r^2(q^2)</a>, Ramanujan J., 20 (2009), 311-328.
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H Y. Yang, <a href="http://dx.doi.org/10.1112/S0024609304003510">Transformation formulas for generalized Dedekind eta functions</a>, Bull. London Math. Soc. 36 (2004), no. 5, 671-682. See p. 679, Table 1.
%F 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.
%F Euler transform of period 10 sequence [ 1, 1, -1, -1, 0, -1, -1, 1, 1, 0, ...].
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u + v)^2 - v * (u^2 - 1).
%F 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))).
%F Convolution inverse of A112274. a(n) = A112274(n) + A132980(n).
%e 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 + ...
%t 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 *)
%o (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))};
%Y Cf. A112274, A132980.
%K sign
%O -1,3
%A _Michael Somos_, Jul 12 2012
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