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A205183 Euler transform of period 5 sequence [ 2, 1, 1, 2, 1, ...].

%I A205183 #24 Feb 16 2025 08:33:16
%S A205183 1,2,4,7,13,21,35,54,84,126,188,273,396,562,794,1106,1533,2099,2862,
%T A205183 3864,5194,6927,9198,12131,15938,20817,27087,35070,45246,58114,74400,
%U A205183 94870,120602,152768,192957,242928,305025,381857,476839,593830,737776
%N A205183 Euler transform of period 5 sequence [ 2, 1, 1, 2, 1, ...].
%C A205183 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A205183 G. C. Greubel, <a href="/A205183/b205183.txt">Table of n, a(n) for n = 0..1000</a>
%H A205183 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A205183 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A205183 Expansion of G(x) / f(-x) = f(-x^2, -x^3) / f(-x)^2 in powers of x where f() is a Ramanujan theta function and G() is a Rogers-Ramanujan function.
%F A205183 G.f. is the limit as n goes to infinity of Sum_{k=0..n} x^k^2 / ((x;x)_k * (x;x)_{n-k}) = Sum_{k=-n..n} (-1)^k * x^(k*(5*k - 1)/2) / ((x;x)_{n-k} * (x;x)_{n+k}).
%F A205183 G.f.: (Sum_{k>=0} x^k^2 / ((1 - x) ... (1 - x^k))) / Product_{k>0} (1 - x^k).
%F A205183 G.f.: (Sum_{k} (-1)^k * x^(k*(5*k - 1)/2)) / Product_{k>0} (1 - x^k)^2.
%F A205183 Convolution of A000041 and A003114.
%F A205183 a(n) ~ exp(Pi*sqrt(14*n/15)) * sqrt(7*phi) / (4*3^(1/2)*5^(3/4)*n), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Mar 24 2018
%e A205183 1 + 2*x + 4*x^2 + 7*x^3 + 13*x^4 + 21*x^5 + 35*x^6 + 54*x^7 + 84*x^8 + ...
%e A205183 q^-7 + 2*q^113 + 4*q^233 + 7*q^353 + 13*q^473 + 21*q^593 + 35*q^713 + ...
%t A205183 f[x_,y_]:= QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; CoefficientList[Series[f[-q^2,-q^3]/QPochhammer[q]^2, {q,0,100}], q] (* _G. C. Greubel_, Apr 14 2018 *)
%o A205183 (PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, 1 / (1 - x^k)^[ 1, 2, 1, 1, 2][k%5 + 1], 1 + x * O(x^n)), n))}
%Y A205183 Cf. A000041, A003114, A113428.
%K A205183 nonn
%O A205183 0,2
%A A205183 _Michael Somos_, Jan 23 2012