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A193883 Expansion of x * phi(x) * psi(x^14) / (f(-x) * f(-x^7)) in powers of x where phi(), psi(), f() are Ramanujan theta functions.

%I A193883 #25 Feb 16 2025 08:33:15
%S A193883 0,1,3,4,7,13,19,29,44,65,94,133,187,258,354,481,651,871,1154,1526,
%T A193883 1998,2603,3376,4358,5594,7148,9103,11531,14560,18320,22972,28708,
%U A193883 35757,44413,54990,67904,83626,102736,125890,153882,187694,228396,277336
%N A193883 Expansion of x * phi(x) * psi(x^14) / (f(-x) * f(-x^7)) in powers of x where phi(), psi(), f() are Ramanujan theta functions.
%C A193883 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A193883 G. C. Greubel, <a href="/A193883/b193883.txt">Table of n, a(n) for n = 0..1000</a>
%H A193883 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A193883 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A193883 Expansion of chi(-x^2)^2 / (chi(-x)^3 * chi(-x^7) * chi(-x^14)^4 ) in powers of x where chi() is a Ramanujan theta function.
%F A193883 Expansion of q^(-5/12) * eta(q^2)^5 * eta(q^28)^2 / (eta(q)^3 * eta(q^4)^2 * eta(q^7) * eta(q^14)) in powers of q.
%F A193883 Euler transform of period 28 sequence [ 3, -2, 3, 0, 3, -2, 4, 0, 3, -2, 3, 0, 3, 0, 3, 0, 3, -2, 3, 0, 4, -2, 3, 0, 3, -2, 3, 0, ...].
%F A193883 a(n) = A102314(4*n + 2).
%F A193883 a(n) ~ exp(4*Pi*sqrt(n/21)) / (2^(5/2) * 21^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Nov 15 2017
%e A193883 G.f. = x + 3*x^2 + 4*x^3 + 7*x^4 + 13*x^5 + 19*x^6 + 29*x^7 + 44*x^8 + 65*x^9 + ...
%e A193883 G.f. = q^17 + 3*q^29 + 4*q^41 + 7*q^53 + 13*q^65 + 19*q^77 + 29*q^89 + 44*q^101 + ...
%t A193883 a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ x^7, x^14], {x, 0, 4 n + 2}];
%t A193883 a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, x^4]^2 / (QPochhammer[ x^7, x^14] QPochhammer[ x^14, x^28]^2 QPochhammer[ x, x^2]^3), {x, 0, n - 1}];
%t A193883 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 2, 0, x^7] / (2 QPochhammer[ x] QPochhammer[ x^7]), {x, 0, n + 3/4}];
%o A193883 (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^28 + A)^2 / (eta(x + A)^3 * eta(x^4 + A)^2 * eta(x^7 + A) * eta(x^14 + A)), n))};
%Y A193883 Cf. A102314.
%K A193883 nonn
%O A193883 0,3
%A A193883 _Michael Somos_, Aug 07 2011