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A201077 G.f.: 1 / Product_{i>=1} (1-q^(2*i-1))^2*(1-q^(12*i-8))*(1-q^(12*i-6))*(1-q^(12*i-4))*(1-q^(12*i)).

%I A201077 #34 Jun 28 2025 06:22:40
%S A201077 1,2,3,6,10,16,26,40,60,90,131,188,269,378,525,726,993,1346,1816,2430,
%T A201077 3230,4274,5619,7348,9570,12400,15994,20554,26303,33530,42602,53934,
%U A201077 68053,85614,107370,134262,167443,208250,258329,319680
%N A201077 G.f.: 1 / Product_{i>=1} (1-q^(2*i-1))^2*(1-q^(12*i-8))*(1-q^(12*i-6))*(1-q^(12*i-4))*(1-q^(12*i)).
%C A201077 The expansion of the denominator is 1, -2, 1, -2, 3, -2, 3, -2, 3, -4, 4, -6, 4, -6, 7, -8, 9, -8, 11, -10, 12, -12, 13, -16, 16, -20, 21,... - _R. J. Mathar_, Nov 27 2011
%C A201077 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A201077 Alois P. Heinz, <a href="/A201077/b201077.txt">Table of n, a(n) for n = 0..1000</a>
%H A201077 Márton Balázs, Dan Fretwell, and Jessica Jay, <a href="https://arxiv.org/abs/2011.05006">Interacting Particle Systems and Jacobi style identities</a>, arXiv:2011.05006 [math.PR], 2020.
%H A201077 Brian Drake, <a href="http://dx.doi.org/10.1016/j.disc.2008.11.020">Limits of areas under lattice paths</a>, Discrete Math. 309 (2009), no. 12, 3936-3953.
%H A201077 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A201077 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A201077 Expansion of chi(x) / (f(-x) * chi(-x^6)) in powers of x where chi(), f() are Ramanujan theta functions. - _Michael Somos_, Jun 07 2012
%F A201077 Expansion of q^(-1/6) * eta(q^2)^2 * eta(q^12) / (eta(q)^2 * eta(q^4) * eta(q^6)) in powers of q. - _Michael Somos_, Jun 07 2012
%F A201077 Expansion of f(x^1, x^5) / (f(-x, -x^2) * f(-x^3, -x^6)) in powers of x where f(, ) is Ramanujan's general theta function. - _Michael Somos_, Feb 18 2017
%F A201077 Euler transform of period 12 sequence [2, 0, 2, 1, 2, 1, 2, 1, 2, 0, 2, 1, ...]. - _Michael Somos_, Feb 18 2017
%F A201077 a(n) ~ exp(2^(3/2)*Pi*sqrt(n)/3) / (3*2^(3/2)*n). - _Vaclav Kotesovec_, Jun 28 2025
%e A201077 G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 10*x^4 + 16*x^5 + 26*x^6 + 40*x^7 + 60*x^8 + ...
%e A201077 G.f. = q + 2*q^7 + 3*q^13 + 6*q^19 + 10*q^25 + 16*q^31 + 26*q^37 + 40*q^43 + ...
%t A201077 max = 39; den[i_] := Series[(1-q^(2*i-1))^2*(1-q^(12*i-8))*(1-q^(12*i-6))*(1-q^(12*i-4))*(1-q^(12*i)), {q, 0, max }] // Normal; gf = 1/Product[den[i], {i, 1, max}]; Series[gf, {q, 0, max}] // CoefficientList[#, q]& (* _Jean-François Alcover_, Mar 18 2014 *)
%t A201077 a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] / (QPochhammer[ x] QPochhammer[ x^6, x^12]), {x, 0, n}]; (* _Michael Somos_, Feb 18 2017 *)
%o A201077 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^12 + A) / (eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A)), n))} /* _Michael Somos_, Jun 07 2012 */
%K A201077 nonn
%O A201077 0,2
%A A201077 _N. J. A. Sloane_, Nov 26 2011