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%I #34 Mar 12 2021 22:24:43
%S 1,0,1,1,0,0,1,1,1,1,1,1,2,1,2,2,2,2,3,2,3,4,3,4,5,4,5,6,6,6,8,7,8,9,
%T 9,10,12,11,13,14,14,15,19,17,20,22,21,23,27,26,29,32,32,34,39,38,43,
%U 46,47,50,56,55,61,67,67,72,80,79,86,93,96,101,112,112,121,130,133
%N Coefficients of replicable function number "126a".
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H Vaclav Kotesovec, <a href="/A112222/b112222.txt">Table of n, a(n) for n = 0..10000</a>
%H 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 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 <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F Expansion of (psi(x^4) * phi(x^42) - x^10 * phi(x^22) * psi(x^84)) / (f(-x^6) * f(-x^14)) = (psi(x^12) * phi(x^14) - x^2 * phi(x^6) * psi(x^28)) / (f(-x^2) * f(-x^42)) in powers of x where phi(), psi(), f() are Ramanujan theta functions [Zhu Cao, May 24 2018]. - _Michael Somos_, May 24 2018
%F Convolution square is A112204, cube is A058675, sixth power is A058563.
%F a(n) ~ exp(2*Pi*sqrt(2*n/7)/3) / (2^(3/4) * sqrt(3) * 7^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, May 30 2018
%F From _G. C. Greubel_, Jul 03 2018: (Start)
%F Expansion of sqrt(T63a) in powers of q, where T63a = A112204.
%F Expansion of (T21A)^(1/6) in powers of q, where T21A = A058563. (End)
%e G.f. = 1 + x^2 + x^3 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + 2*x^12 + x^13 + ...
%e T126a = 1/q + q^11 + q^17 + q^35 + q^41 + q^47 + q^53 + q^59 + q^65 +...
%t a[ n_] := With[ {A = (QPochhammer[ x^3] QPochhammer[ x^7] / (QPochhammer[ x] QPochhammer[ x^21]))^2}, SeriesCoefficient[ (A - 2 x + x^2/A)^(1/6), {x, 0, n}]]; (* _Michael Somos_, May 24 2018 *)
%t a[ n_] := With[ {A = QPochhammer[ x] QPochhammer[ x^3] / (QPochhammer[ x^7] QPochhammer[ x^21])}, SeriesCoefficient[ (A + x + 7 x^2/A)^(1/6), {x, 0, n}]]; (* _Michael Somos_, May 24 2018 *)
%t a[ n_] := SeriesCoefficient[ (1/2) x^(-3/4) (EllipticTheta[ 2, 0, x^3] EllipticTheta[ 3, 0, x^7] - EllipticTheta[ 3, 0, x^3] EllipticTheta[ 2, 0, x^7]) / (QPochhammer[ x] QPochhammer[ x^21]), {x, 0, n}, Assumptions -> x > 0]; (* _Michael Somos_, May 24 2018 *)
%t a[ n_] := SeriesCoefficient[ (1/2) x^(-1/4) (EllipticTheta[ 2, 0, x^1] EllipticTheta[ 3, 0, x^21] - EllipticTheta[ 3, 0, x^1] EllipticTheta[ 2, 0, x^21]) /(QPochhammer[ x^3] QPochhammer[ x^7]), {x, 0, n}, Assumptions -> x > 0]; (* _Michael Somos_, May 24 2018 *)
%t CoefficientList[Series[((QPochhammer[x^3]^2 * QPochhammer[x^7]^2 - x*QPochhammer[x]^2 * QPochhammer[x^21]^2) / (QPochhammer[x] * QPochhammer[x^3] * QPochhammer[x^7] * QPochhammer[x^21]))^(1/3), {x, 0, 100}], x] (* _Vaclav Kotesovec_, May 30 2018 *)
%t eta[q_] := q^(1/24)*QPochhammer[q]; nmax = 120; b:= eta[q]*eta[q^3]/ (eta[q^7]*eta[q^21]); T21A := 1 + b + 7/b; T126a := CoefficientList[ Series[(q*T21A + O[q]^nmax)^(1/6), {q, 0, 100}], q]; Table[T126a[[n]], {n, 1, 80}] (* _G. C. Greubel_, Jul 03 2018 *)
%o (PARI) q='q+O('q^80); b = eta(q)*eta(q^3)/(q*eta(q^7)*eta(q^21)); T21A = b + 1 + 7/b; Vec((q*T21A)^(1/6)) \\ _G. C. Greubel_, Jul 03 2018
%Y Cf. A058563, A058675, A112204.
%K nonn
%O 0,13
%A _Michael Somos_, Aug 28 2005
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