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%I #13 Mar 12 2021 22:24:44
%S 1,1,1,2,3,4,6,8,11,15,19,25,33,42,53,68,86,107,134,166,205,253,309,
%T 377,460,557,672,811,974,1166,1394,1661,1975,2344,2773,3275,3863,4543,
%U 5333,6253,7316,8544,9964,11600,13484,15653,18140,20994,24269,28011,32288
%N Expansion of psi(x^6) / psi(-x) in powers of x where psi() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A132217/b132217.txt">Table of n, a(n) for n = 0..1000</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>
%F Expansion of q^(-5/8) * eta(q^2) * eta(q^12)^2 / (eta(q) * eta(q^4) * eta(q^6)) in powers of q.
%F Euler transform of period 12 sequence [ 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, ...].
%F Product_{k>0} (1 - x^(12*k)) * (1 - x^(2*k) + x^(4*k)) / (1 - x^k).
%F Expansion of f(-x^2, -x^10) / f(-x, -x^2) in powers of x where f(, ) is Ramanujan's general theta function. - _Michael Somos_, Oct 06 2015
%F Number of partitions of n into parts not congruent to 0, 2, 10 (mod 12). - _Michael Somos_, Oct 06 2015
%F a(2*n) = A262987(n). - _Michael Somos_, Oct 06 2015
%F a(n) ~ exp(sqrt(n/2)*Pi) / (2^(11/4) * sqrt(3) * n^(3/4)). - _Vaclav Kotesovec_, Oct 06 2015
%e G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 11*x^8 + 15*x^9 + ...
%e G.f. = q^5 + q^13 + q^21 + 2*q^29 + 3*q^37 + 4*q^45 + 6*q^53 + 8*q^61 + 11*q^69 + ...
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^3] / (2^(1/2) x^(5/8) EllipticTheta[ 2, Pi/4, x^(1/2)]), {x, 0, n}]; (* _Michael Somos_, Oct 06 2015 *)
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^12] QPochhammer[ x^2, x^12] QPochhammer[ x^10, x^12] / QPochhammer[ x], {x, 0, n}]; (* _Michael Somos_, Oct 06 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff(eta(x^2 + A) * eta(x^12 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)), n))};
%Y Cf. A262987.
%K nonn
%O 0,4
%A _Michael Somos_, Aug 13 2007
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