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%I #12 Mar 12 2021 22:24:45
%S 1,8,39,152,513,1560,4382,11552,28899,69168,159372,355224,768885,
%T 1621296,3339201,6732232,13311450,25854744,49398043,92953016,
%U 172451760,315744072,570997539,1020691248,1804730732,3158323272,5473566645,9398873032,15998363307,27005721648
%N Expansion of phi(x) / f(-x)^6 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="/A134415/b134415.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 Expansion of q^(1/4) * eta(q^2)^5 / (eta(q)^8 * eta(q^4)^2) in powers of q.
%F Euler transform of period 4 sequence [ 8, 3, 8, 5, ...].
%F G.f.: Product_{k>0} (1 + x^k)^3 / ((1 - x^k)^5 * (1 + x^(2*k))^2).
%F a(n) = A134414(4*n - 1).
%F a(n) ~ exp(2*Pi*sqrt(n)) / (16*n^2). - _Vaclav Kotesovec_, Sep 08 2015
%F Convolution inverse of A244276. - _Michael Somos_, Oct 25 2015
%e G.f. = 1 + 8*x + 39*x^2 + 152*x^3 + 513*x^4 + 1560*x^5 + 4382*x^6 + 11552*x^7 + ...
%e G.f. = 1/q + 8*q^3 + 39*q^7 + 152*q^11 + 513*q^15 + 1560*q^19 + 4382*q^23 + ...
%t nmax = 40; CoefficientList[Series[Product[(1 + x^k)^3 / ((1 - x^k)^5 * (1 + x^(2*k))^2), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 08 2015 *)
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] / QPochhammer[ x]^6, {x, 0, n}]; (* _Michael Somos_, Oct 25 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 / (eta(x + A)^8 * eta(x^4 + A)^2), n))};
%Y Cf. A134414.
%K nonn
%O 0,2
%A _Michael Somos_, Oct 26 2007
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