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A260600 Expansion of x * psi(x^3) * psi(x^12) / f(-x) in powers of x where psi(), f() are Ramanujan theta functions.

%I A260600 #33 Feb 16 2025 08:33:26
%S A260600 0,1,1,2,4,6,9,14,20,29,42,58,80,111,149,200,268,353,463,606,784,1011,
%T A260600 1299,1656,2104,2664,3354,4208,5264,6555,8138,10076,12428,15288,18758,
%U A260600 22944,27996,34081,41377,50124,60592,73075,87951,105652,126652,151547,181015
%N A260600 Expansion of x * psi(x^3) * psi(x^12) / f(-x) in powers of x where psi(), f() are Ramanujan theta functions.
%C A260600 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A260600 G. C. Greubel, <a href="/A260600/b260600.txt">Table of n, a(n) for n = 0..1000</a>
%H A260600 Vaclav Kotesovec, <a href="https://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], 2015-2016.
%H A260600 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A260600 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A260600 Expansion of q^(-5/6) * eta(q^6)^2 * eta(q^24)^2 / (eta(q) * eta(q^3) * eta(q^12)) in powers of q.
%F A260600 Euler transform of period 24 sequence [1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, -1, ...].
%F A260600 -2 * a(n) = A260574(4*n + 3).
%F A260600 a(n) ~ exp(sqrt(2*n/3)*Pi) / (24*sqrt(2*n)). - _Vaclav Kotesovec_, Oct 14 2015
%e A260600 G.f. = x + x^2 + 2*x^3 + 4*x^4 + 6*x^5 + 9*x^6 + 14*x^7 + 20*x^8 + ...
%e A260600 G.f. = q^11 + q^17 + 2*q^23 + 4*q^29 + 6*q^35 + 9*q^41 + 14*q^47 + ...
%t A260600 a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(3/2)] EllipticTheta[ 2, 0, x^6] / ( 4 x^(7/8) QPochhammer[ x]), {x, 0, n}];
%t A260600 nmax=60; CoefficientList[Series[x*Product[(1-x^(6*k)) * (1-x^(24*k)) * (1+x^(3*k)) * (1+x^(12*k)) / ((1-x^k)),{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Oct 14 2015 *)
%o A260600 (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^6 + A)^2 * eta(x^24 + A)^2 / (eta(x + A) * eta(x^3 + A) * eta(x^12 + A)), n))};
%o A260600 (PARI) q='q+O('q^99); concat(0, Vec(eta(q^6)^2*eta(q^24)^2 / (eta(q)*eta(q^3)*eta(q^12)))) \\ _Altug Alkan_, Mar 18 2018
%Y A260600 Cf. A260574.
%K A260600 nonn
%O A260600 0,4
%A A260600 _Michael Somos_, Jul 29 2015