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A112206 Coefficients of replicable function number "72b".

%I A112206 #32 May 06 2025 11:15:55
%S A112206 1,1,0,2,2,1,2,2,3,4,4,4,7,7,6,10,11,11,14,16,17,21,22,24,32,34,34,44,
%T A112206 49,50,60,66,72,84,90,98,117,125,132,156,171,181,206,226,245,277,298,
%U A112206 322,369,397,422,480,522,557,620,674,728,807,868,936,1043,1121,1198
%N A112206 Coefficients of replicable function number "72b".
%C A112206 From _Michael Somos_, Oct 28 2019: (Start)
%C A112206 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A112206 Convolution squared is A112173.
%C A112206 G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = f(t) where q = exp(2 Pi i t).
%C A112206 Given G.f. A(x), then B(q) = q^(-1) * A(q^6) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = 2 + (u^2 - v)*v*w^2 + (u^2 + v)*v^2.
%C A112206 (End)
%H A112206 G. C. Greubel, <a href="/A112206/b112206.txt">Table of n, a(n) for n = 0..1000</a>
%H A112206 D. Ford, J. McKay and S. P. Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%H A112206 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A112206 <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F A112206 a(n) ~ exp(sqrt(2*n)*Pi/3) / (2^(5/4) * sqrt(3) * n^(3/4)). - _Vaclav Kotesovec_, Sep 08 2015
%F A112206 Expansion of  q^(1/6)*((eta(q^2)*eta(q^6))^2/(eta(q)*eta(q^3)*eta(q^4) *eta(q^12))) in powers of q. - _G. C. Greubel_, Jun 01 2018
%F A112206 From _Michael Somos_, Oct 28 2019: (Start)
%F A112206 Expansion of chi(x) * chi(x^3) in powers of x where chi() is a Ramanujan theta function.
%F A112206 Euler transform of period 12 sequence [1, -1, 2, 0, 1, -2, 1, 0, 2, -1, 1, 0, ...].
%F A112206 G.f.: Product_{k>=0} (1 + x^(2*k + 1)) * (1 + x^(6*k + 3)).
%F A112206 a(n) = (-1)^n * A112175(n). a(2*n) = A328789(n). a(2*n + 1) = A328790(n).
%F A112206 (End)
%e A112206 G.f. = 1 + x + 2*x^3 + 2*x^4 + x^5 + 2*x^6 + 2*x^7 + 3*x^8 + ...
%e A112206 G.f. = q^-1 + q^5 + 2*q^17 + 2*q^23 + q^29 + 2*q^35 + 2*q^41 + ...
%t A112206 nmax = 60; CoefficientList[Series[Product[(1 + x^k)*(1 + x^(3*k)) / ((1 + x^(2*k))*(1 + x^(6*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 08 2015 *)
%t A112206 eta[q_]:= q^(1/24)*QPochhammer[q]; h:= q^(1/6)*((eta[q^2]*eta[q^6])^2/(eta[q]*eta[q^3]*eta[q^4]*eta[q^12])); a:= CoefficientList[Series [h, {q,0,60}], q]; Table[a[[n]], {n,1,50}] (* _G. C. Greubel_, Jun 01 2018 *)
%t A112206 a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^3, x^6], {x, 0 ,n}]; (* _Michael Somos_, Oct 28 2019 *)
%o A112206 (PARI) q='q+O('q^50); h=((eta(q^2)*eta(q^6))^2/(eta(q)*eta(q^3)*eta(q^4) *eta(q^12))); Vec(h) \\ _G. C. Greubel_, Jun 01 2018
%o A112206 (PARI) {a(n) = my(A); if( n < 0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A))^2 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A)), n))}; /* _Michael Somos_, Oct 28 2019 */
%Y A112206 Cf. A112173, A112175, A328789, A328790.
%K A112206 nonn
%O A112206 0,4
%A A112206 _Michael Somos_, Aug 28 2005