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A193826 Expansion of psi(x^2) * phi(x^7) / (f(-x) * f(-x^7)) in powers of x where phi(), psi(), f() are Ramanujan theta functions.

%I A193826 #23 Feb 16 2025 08:33:15
%S A193826 1,1,3,4,7,10,17,26,38,57,81,114,161,224,309,419,569,759,1011,1336,
%T A193826 1757,2296,2981,3855,4956,6344,8087,10272,12994,16367,20561,25723,
%U A193826 32086,39902,49484,61182,75439,92791,113821,139294,170073,207187,251853
%N A193826 Expansion of psi(x^2) * phi(x^7) / (f(-x) * f(-x^7)) in powers of x where phi(), psi(), f() are Ramanujan theta functions.
%C A193826 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A193826 G. C. Greubel, <a href="/A193826/b193826.txt">Table of n, a(n) for n = 0..1000</a>
%H A193826 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A193826 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A193826 Expansion of chi(-x^14)^2 / (chi(-x) * chi(-x^2)^4 * chi(-x^7)^3 ) in powers of x where chi() is a Ramanujan theta function.
%F A193826 Expansion of q^(1/12) * eta(q^4)^2 * eta(q^14)^5 / (eta(q) * eta(q^2) * eta(q^7)^3* eta(q^28)^2) in powers of q.
%F A193826 Euler transform of period 28 sequence [ 1, 2, 1, 0, 1, 2, 4, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 4, 2, 1, 0, 1, 2, 1, 0, ...].
%F A193826 a(n) = A102314(4*n).
%F A193826 a(n) ~ exp(4*Pi*sqrt(n/21)) / (2^(5/2) * 21^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Nov 15 2017
%e A193826 G.f. = 1 + x + 3*x^2 + 4*x^3 + 7*x^4 + 10*x^5 + 17*x^6 + 26*x^7 + 38*x^8 + ...
%e A193826 G.f. = 1/q + q^11 + 3*q^23 + 4*q^35 + 7*q^47 + 10*q^59 + 17*q^71 + 26*q^83 + ...
%t A193826 a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ x^7, x^14], {x, 0, 4 n}];
%t A193826 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^7] EllipticTheta[ 2, 0, x] / (2 x^(1/4) QPochhammer[ x] QPochhammer[ x^7]), {x, 0, n}];
%t A193826 a[ n_] := SeriesCoefficient[ QPochhammer[ x^14, x^28]^2 / (QPochhammer[ x, x^2] QPochhammer[ x^2, x^4]^2 QPochhammer[ x^7, x^14]^3), {x, 0, n}];
%o A193826 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^2 * eta(x^14 + A)^5 / (eta(x + A) * eta(x^2 + A) * eta(x^7 + A)^3* eta(x^28 + A)^2), n))};
%Y A193826 Cf. A102314.
%K A193826 nonn
%O A193826 0,3
%A A193826 _Michael Somos_, Aug 06 2011