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%I #19 Mar 12 2021 22:24:44
%S 1,3,6,13,24,42,73,123,201,320,504,774,1172,1755,2592,3789,5478,7851,
%T 11146,15696,21942,30456,42000,57546,78403,106212,143124,191925,
%U 256146,340320,450204,593163,778416,1017698,1325784,1721157,2227050,2872422
%N Expansion of ((eta(q^2) * eta(q^14)) / (eta(q) * eta(q^7)))^3 in powers of q.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A120006/b120006.txt">Table of n, a(n) for n = 1..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 * (chi(-q) * chi(-q^7))^3 in powers of q where chi() is a Ramanujan theta function.
%F Euler transform of period 14 sequence [ 3, 0, 3, 0, 3, 0, 6, 0, 3, 0, 3, 0, 3, 0, ...].
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v - 6*u*v - 8*u*v^2.
%F G.f.: x * (Product_{k>0} (1 + x^k) * (1 + x^(7*k)))^3.
%F Convolution inverse of A132319.
%F a(n) ~ exp(2*Pi*sqrt(2*n/7)) / (8 * 2^(3/4) * 7^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2015
%e q + 3*q^2 + 6*q^3 + 13*q^4 + 24*q^5 + 42*q^6 + 73*q^7 + 123*q^8 + 201*q^9 + ...
%t nmax = 40; Rest[CoefficientList[Series[x * Product[((1 + x^k) * (1 + x^(7*k)))^3, {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Sep 07 2015 *)
%t eta[q_] := q^(1/24)*QPochhammer[q]; Rest[CoefficientList[Series[(( eta[q^2]*eta[q^14])/(eta[q]*eta[q^7]))^3, {q, 0, 50}], q]] (* _G. C. Greubel_, Apr 19 2018 *)
%o (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^14 + A) / (eta(x + A) * eta(x^7 + A)))^3, n))}
%Y Cf. A132319.
%K nonn
%O 1,2
%A _Michael Somos_, Jun 02 2006
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