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%I #32 Mar 12 2021 22:24:42
%S 1,9,27,82,207,486,1055,2205,4374,8427,15696,28539,50630,88119,150417,
%T 252727,418068,682344,1099343,1750968,2758185,4301682,6645150,
%U 10175625,15451744,23281686,34819227,51712860,76292784,111850740,162997314
%N Number of 3-colored generalized Frobenius partitions of n.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A053762/b053762.txt">Table of n, a(n) for n = 0..1000</a>
%H G. E. Andrews, <a href="https://bookstore.ams.org/memo-49-301/">Generalized Frobenius Partitions</a>, AMS Memoir 301, 1984 (sequence is denoted c\phi_3(n)).
%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^(1/8) * (eta(q)^3 + 9 * eta(q^9)^3) / (eta(q)^3 * eta(q^3)) in powers of q. - _Michael Somos_, Mar 09 2011
%F Expansion of a(x) / f(-x)^3 in powers of x where a() is a cubic AGM theta function and f() is a Ramanujan theta function. - _Michael Somos_, Aug 21 2012
%F Convolution of A000716 and A004016. - _Michael Somos_, Mar 09 2011
%F a(n) ~ exp(sqrt(2*n)*Pi)/(4*sqrt(3)*n). - _Vaclav Kotesovec_, Nov 13 2016
%e 1 + 9*x + 27*x^2 + 82*x^3 + 207*x^4 + 486*x^5 + 1055*x^6 + 2205*x^7 + ...
%e 1/q + 9*q^7 + 27*q^15 + 82*q^23 + 207*q^31 + 486*q^39 + 1055*q^47 + 2205*q^55 + ...
%t nmax = 30; CoefficientList[Series[(Product[(1 - x^k)^3, {k, 1, nmax}] + 9*x*Product[(1 - x^(9*k))^3, {k, 1, nmax}]) / Product[((1 - x^k)^3*(1 - x^(3*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 13 2016 *)
%t a[n_]:= SeriesCoefficient[q^(1/8)*(eta[q]^3 + 9*eta[q^9]^3)/(eta[q]^3* eta[q^3]), {q, 0, n}]; Table[a[n], {n,0,50}] (* _G. C. Greubel_, Feb 08 2018 *)
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3 + 9 * x * eta(x^9 + A)^3) / (eta(x + A)^3 * eta(x^3 + A)), n))} /* _Michael Somos_, Mar 09 2011 */
%Y Cf. A000716, A004016, A051136.
%K easy,nonn
%O 0,2
%A _James A. Sellers_, Apr 05 2000
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