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%I #28 Mar 12 2021 22:24:42
%S 1,5,27,41,146,243,510,887,1755,2728,5052,7857,13157,20253,32805,
%T 48680,76568,112320,169814,246263,365013,519046,755632,1063368,
%U 1516404,2112551,2972160,4089098,5683166,7750782,10633276,14382932,19539387,26192432,35263852
%N Coefficients of replicable function number 12b.
%C The convolution square of this sequence is A007254 except for the constant term: T12b(q)^2 = T6A(q^2) + 10. - _G. A. Edgar_, Apr 15 2017
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H Vaclav Kotesovec, <a href="/A058490/b058490.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..502 from G. A. Edgar)
%H D. Alexander, C. Cummins, J. McKay and C. Simons, <a href="http://oeis.org/A007242/a007242_1.pdf">Completely Replicable Functions</a>, LMS Lecture Notes, 165, ed. Liebeck and Saxl (1992), 87-98, annotated and scanned copy.
%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="/A007191/a007191.pdf">Emails to N. J. A. Sloane, 1993</a>
%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 q^(1/2) * (eta(q)^3*eta(q^3)^3 / (eta(q^2)^3*eta(q^6)^3) + 8 *eta(q^2)^3*eta(q^6)^3 / (eta(q)^3*eta(q^3)^3)) in powers of q. - _G. A. Edgar_, Apr 15 2017
%F From _Michael Somos_, Jun 12 2017: (Start)
%F Expansion of (chi(-x) * chi(-x^3))^3 + 8*x/(chi(-x) * chi(-x^3))^3 = (chi(-x^3) / chi(-x))^6 - x*(chi(-x) / chi(-x^3))^6 in powers of x.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = f(t) where q = exp(2 Pi i t).
%F Convolution square is A288630.
%F a(n) = 2*A058484(n) - A058206(n) = 2*A058492(n) - A058489(n). (End)
%F a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 13 2017
%e T12b = 1/q + 5*q + 27*q^3 + 41*q^5 + 146*q^7 + 243*q^9 + 510*q^11 + ...
%t a[ n_] := With[{A = (QPochhammer[ x^2] QPochhammer[ x^3] / (QPochhammer[ x] QPochhammer[ x^6]))^6}, SeriesCoefficient[ A - x / A, {x, 0, n}]]; (* _Michael Somos_, Jun 12 2017 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); A = (eta(x^2 + A) * eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)))^6; polcoeff( A - x/A, n))}; /* _Michael Somos_, Jun 12 2017 */
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); A = (eta(x + A) * eta(x^3 + A) / (eta(x^2 + A) * eta(x^6 + A)))^3; polcoeff( A + 8*x/A, n))}; /* _Michael Somos_, Jun 12 2017 */
%Y Cf. A000521, A007240, A007254, A014708, A007241, A007267, A045478, etc.
%Y Cf. A058484, A058492, A156215, A288630.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Nov 27 2000
%E More terms from _Michael Somos_, Feb 06 2009
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