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A146163 Expansion of q^(-3/4) * eta(q^2)^2 * eta(q^20) / (eta(q)^2 * eta(q^4)) in powers of q.

%I A146163 #13 Oct 03 2024 23:29:07
%S A146163 1,2,3,6,10,16,25,38,57,84,121,172,243,338,465,636,862,1158,1546,2050,
%T A146163 2701,3540,4613,5980,7719,9916,12682,16158,20506,25926,32667,41022,
%U A146163 51348,64080,79730,98922,122407,151068,185968,228384,279816,342052
%N A146163 Expansion of q^(-3/4) * eta(q^2)^2 * eta(q^20) / (eta(q)^2 * eta(q^4)) in powers of q.
%H A146163 G. C. Greubel, <a href="/A146163/b146163.txt">Table of n, a(n) for n = 0..1000</a>
%F A146163 Euler transform of period 20 sequence [ 2, 0, 2, 1, 2, 0, 2, 1, 2, 0, 2, 1, 2, 0, 2, 1, 2, 0, 2, 0, ...].
%F A146163 a(n) ~ exp(2*Pi*sqrt(n/5)) / (4*5^(3/4)*n^(3/4)). - _Vaclav Kotesovec_, Jul 11 2016
%F A146163 a(n) = A146162(4*n + 3).
%e A146163 q^3 + 2*q^7 + 3*q^11 + 6*q^15 + 10*q^19 + 16*q^23 + 25*q^27 + 38*q^31 + ...
%t A146163 nmax = 50; CoefficientList[Series[Product[(1+x^k)^2 * (1-x^(20*k)) / (1-x^(4*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jul 11 2016 *)
%t A146163 a[n_]:= SeriesCoefficient[QPochhammer[-q, q]^2*QPochhammer[q^20, q^20]/(QPochhammer[q^4, q^4]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Dec 05 2017 *)
%o A146163 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^20 + A) / (eta(x + A)^2 * eta(x^4 + A)), n))}
%K A146163 nonn
%O A146163 0,2
%A A146163 _Michael Somos_, Oct 27 2008