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A085140 Expansion of q^(-1/6) * eta(q^2)^3 / eta(q)^2 in powers of q.

%I A085140 #33 Feb 16 2025 08:32:50
%S A085140 1,2,2,4,5,6,10,12,15,20,26,32,40,50,60,76,92,110,134,160,191,230,272,
%T A085140 320,380,446,522,612,715,830,966,1120,1292,1494,1720,1976,2272,2602,
%U A085140 2974,3400,3876,4412,5020,5700,6460,7322,8282,9352,10559,11900,13396
%N A085140 Expansion of q^(-1/6) * eta(q^2)^3 / eta(q)^2 in powers of q.
%C A085140 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A085140 In the notation of Dragonette on page 498 Lemma 6, the generating function is G_2(q^(1/2))/2.
%C A085140 Equals A000009 convolved with A010054. [_Gary W. Adamson_, Mar 16 2010]
%H A085140 Seiichi Manyama, <a href="/A085140/b085140.txt">Table of n, a(n) for n = 0..10000</a>
%H A085140 L. A. Dragonette, <a href="http://dx.doi.org/10.1090/S0002-9947-1952-0049927-8">Some Asymptotic Formulae for the Mock Theta Series of Ramanujan</a>, Trans. Amer. Math. Soc., 72 (1952), 474-500.
%H A085140 Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 16.
%H A085140 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A085140 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A085140 Expansion of psi(x) / chi(-x) = f(-x^2) / chi(-x)^2 = f(-x) / chi(-x)^3 = phi(-x) / chi(-x)^4 = phi(x) / chi(-x^2)^2 = f(-x^2)^2 / phi(-x) = f(-x)^4 / phi(-x)^3 = psi(x)^2 / f(-x^2) = chi(x)^2 * psi(x^2) = f(-x^2)^3 / f(-x)^2 in powers of x where f(), phi(), psi(), chi() are Ramanujan theta functions. - _Michael Somos_, Feb 18 2006
%F A085140 Euler transform of period 2 sequence [ 2, -1, ...].
%F A085140 G.f.: Product_{k>0} (1 - x^(2*k)) / (1 - x^(2*k - 1))^2 = Product_{k>0} (1 - x^k) * (1 + x^k)^3.
%F A085140 a(n) = b(n)+b(n-1)+b(n-3)+b(n-6)+...+b(n-k*(k+1)/2)+..., where b() is A000009(). E.g. a(8) = b(8)+b(7)+b(5)+b(2) = 6+5+3+1 = 15. - _Vladeta Jovovic_, Aug 18 2004
%F A085140 G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = (3/4)^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132970. - _Michael Somos_, Jul 11 2015
%F A085140 a(n) = A053254(2*n). - _Michael Somos_, Jul 11 2015
%F A085140 a(n) ~ exp(Pi*sqrt(n/3))/(4*sqrt(n)). - _Vaclav Kotesovec_, Sep 07 2015
%e A085140 G.f. = 1 + 2*x + 2*x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 10*x^6 + 12*x^7 + 15*x^8 + ...
%e A085140 G.f. = q + 2*q^7 + 2*q^13 + 4*q^19 + 5*q^25 + 6*q^31 + 10*q^37 + 12*q^43 + 15*q^49 + ...
%t A085140 a[ n_] := SeriesCoefficient[ Product[ (1 - x^k) * (1 + x^k)^3, {k, n}], {x, 0, n}];
%t A085140 a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, 2, n, 2}] / Product[ 1 - x^k, {k, 1, n, 2}]^2, {x, 0, n}];
%t A085140 a[ n_] := With[ {t = Log[q]/(2 Pi I)}, SeriesCoefficient[ q^(-1/6) DedekindEta[ 2 t]^3 / DedekindEta[ t]^2, {q, 0, n}]];
%t A085140 a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ -x, x]^3, {x, 0, n}]; (* _Michael Somos_, Jul 11 2015 *)
%t A085140 a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^(j j + j) / Product[ 1 + x^k, {k, 1, 2 j + 1, 2}], {j, 0, Sqrt[8 n + 1]/2}], {x, 0, 2 n}]]; (* _Michael Somos_, Jul 11 2015 *)
%o A085140 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 / eta(x + A)^2, n))};
%Y A085140 Cf. A000009, A010054. [_Gary W. Adamson_, Mar 16 2010]
%Y A085140 Cf. A053254, A132970.
%K A085140 nonn
%O A085140 0,2
%A A085140 _Michael Somos_, Jun 20 2003