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A123653 Expansion of (eta(q^2)eta(q^6)/(eta(q)eta(q^3)))^6 in powers of q.

%I A123653 #20 Feb 16 2025 08:33:03
%S A123653 1,6,21,68,198,510,1248,2904,6393,13604,28044,55956,108982,207552,
%T A123653 386622,707216,1271970,2250582,3925780,6757272,11483232,19290824,
%U A123653 32057352,52722744,85884503,138644292,221885805,352241792,554892894
%N A123653 Expansion of (eta(q^2)eta(q^6)/(eta(q)eta(q^3)))^6 in powers of q.
%C A123653 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).
%C A123653 Expansion of q/(chi(-q)*chi(-q^3))^6 in powers of q where chi() is a Ramanujan theta function.
%H A123653 Seiichi Manyama, <a href="/A123653/b123653.txt">Table of n, a(n) for n = 1..10000</a>
%H A123653 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A123653 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A123653 Euler transform of period 6 sequence [ 6, 0, 12, 0, 6, 0, ...].
%F A123653 G.f. A(x) satisfies 0=f(A(x), A(x^2)) where f(u, v)= u^2 -v*(1+12*u+64*u*v)
%F A123653 G.f.: x*(Product_{k>0} (1+x^k)*(1+x^(3k)))^6.
%F A123653 a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (64 * 2^(3/4) * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2015
%F A123653 Convolution inverse of A121666. - _Seiichi Manyama_, Mar 30 2017
%t A123653 nmax = 40; Rest[CoefficientList[Series[x * Product[((1+x^k) * (1+x^(3*k)))^6, {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Sep 07 2015 *)
%o A123653 (PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( (eta(x^2+A)*eta(x^6+A)/eta(x+A)/eta(x^3+A))^6, n))}
%K A123653 nonn
%O A123653 1,2
%A A123653 _Michael Somos_, Oct 04 2006