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%I #12 Mar 12 2021 22:24:44
%S 1,2,2,4,6,8,12,18,26,34,48,64,84,112,146,192,246,316,402,508,640,804,
%T 1008,1248,1548,1910,2344,2872,3510,4276,5184,6280,7578,9120,10956,
%U 13128,15702,18724,22292,26480,31392,37148,43884,51760,60912,71592
%N Expansion of f(q)*f(q^7)/(f(-q)*f(-q^7)) in powers of q where f() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
%H G. C. Greubel, <a href="/A123862/b123862.txt">Table of n, a(n) for n = 0..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 Euler transform of period 28 sequence [ 2, -1, 2, 0, 2, -1, 4, 0, 2, -1, 2, 0, 2, -2, 2, 0, 2, -1, 2, 0, 4, -1, 2, 0, 2, -1, 2, 0, ...].
%F G.f. A(x) satisfies 0=f(A(x), A(x^2)) where f(u, v)=(u-1)^2 -2*u*v*(v-1).
%F a(n) ~ exp(2*Pi*sqrt(n/7)) / (4 * 7^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 03 2018
%t QP := QPochhammer; a[n_]:= SeriesCoefficient[QP[-q]*QP[-q^7]/( QP[q]* QP[q^7]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Jan 04 2018 *)
%o (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^2+A)*eta(x^14+A))^3/ (eta(x+A)*eta(x^7+A))^2/ (eta(x^4+A)*eta(x^28+A)), n))}
%Y Cf. A123648(n)=a(n)/2 if n>0.
%K nonn
%O 0,2
%A _Michael Somos_, Oct 14 2006
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