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 A260414 Expansion of psi(x^3) * psi(x^6) / f(-x^4) in powers of x where phi(), f() are Ramanujan theta functions. 1

%I

%S 1,0,0,1,1,0,1,1,2,2,1,2,3,2,2,4,5,4,5,6,7,7,7,9,12,11,11,15,16,16,18,

%T 21,24,25,26,31,36,35,38,45,50,51,55,63,69,73,77,87,98,101,107,122,

%U 132,138,149,164,180,190,201,222,243,254,271,300,324,340,364

%N Expansion of psi(x^3) * psi(x^6) / f(-x^4) in powers of x where phi(), f() are Ramanujan theta functions.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H G. C. Greubel, <a href="/A260414/b260414.txt">Table of n, a(n) for n = 0..1000</a>

%H 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 M. 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^(-23/24) * eta(q^6) * eta(q^12)^2 / (eta(q^3) * eta(q^4)) in powers of q.

%F Euler transform of period 12 sequence [ 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, -1, ...].

%F a(n) ~ exp(Pi*sqrt(n/6)) / (12*sqrt(n)). - _Vaclav Kotesovec_, Jul 11 2016

%e G.f. = 1 + x^3 + x^4 + x^6 + x^7 + 2*x^8 + 2*x^9 + x^10 + 2*x^11 + ...

%e G.f. = q^23 + q^95 + q^119 + q^167 + q^191 + 2*q^215 + 2*q^239 + q^263 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(3/2)] EllipticTheta[ 2, 0, x^3] / (4 x^(9/8) QPochhammer[ x^4]), {x, 0, n}];

%t nmax = 100; CoefficientList[Series[Product[(1+x^(3*k)) * (1-x^(12*k))^2 / (1-x^(4*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jul 11 2016 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A) * eta(x^12 + A)^2 / (eta(x^3 + A) * eta(x^4 + A)), n))};

%K nonn

%O 0,9

%A _Michael Somos_, Jul 24 2015

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Last modified April 6 15:41 EDT 2020. Contains 333276 sequences. (Running on oeis4.)