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A029552 Expansion of phi(x) / f(-x) in powers of x where phi(), f() are Ramanujan theta functions.

%I A029552 #30 Feb 16 2025 08:32:35
%S A029552 1,3,4,7,13,19,29,43,62,90,126,174,239,325,435,580,769,1007,1313,1702,
%T A029552 2191,2808,3580,4539,5735,7216,9036,11278,14028,17383,21474,26448,
%U A029552 32471,39759,48550,59123,71829,87053,105249,126975,152858,183623
%N A029552 Expansion of phi(x) / f(-x) in powers of x where phi(), f() are Ramanujan theta functions.
%C A029552 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A029552 G. C. Greubel, <a href="/A029552/b029552.txt">Table of n, a(n) for n = 0..1000</a>
%H A029552 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A029552 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A029552 Expansion of q^(1/24) * eta(q^2)^5 /(eta(q)^3 * eta(q^4)^2) in powers of q. - _Michael Somos_, Sep 17 2004
%F A029552 Euler transform of period 4 sequence [3, -2, 3, 0, ...]. - _Michael Somos_, Sep 17 2004
%F A029552 G.f. A(x) is the limit of x^(n^2) P_{2n}(1/x) where P_n(q) = Sum_{k=0..n} C(n,k;q) and C(n,k;q) is q-binomial coefficients. See A083906 for P_n. - _Michael Somos_, Sep 17 2004
%F A029552 G.f.: (1 + 2 * Sum_{k>0} x^(k^2)) / (Product_{k>0} (1 - x^k)).
%F A029552 a(n) ~ exp(sqrt(2*n/3)*Pi) / (2^(7/4)*3^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, May 01 2017
%F A029552 Expansion of chi(x)^3/chi(-x^2) = chi(x)^2/chi(-x) = chi(-x^2)^2/chi(-x)^3 in powers of x where chi() is a Ramanujan theta function. - _Michael Somos_, Apr 24 2023
%e A029552 G.f. = 1 + 3*x + 4*x^2 + 7*x^3 + 13*x^4 + 19*x^5 + 29*x^6 + 43*x^7 + ...
%e A029552 G.f. = 1/q + 3*q^23 + 4*q^47 + 7*q^71 + 13*q^95 + 19*q^119 + 29*q^143 + ...
%t A029552 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] / QPochhammer[ q], {q, 0, n}]; (* _Michael Somos_, Oct 29 2013 *)
%t A029552 a[ n_] := SeriesCoefficient[ QPochhammer[ -q, q^2]^2 / QPochhammer[ q, q^2], {q, 0, n}]; (* _Michael Somos_, Oct 29 2013 *)
%o A029552 (PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=1, sqrtint(n), 2*x^k^2, 1) / eta(x + x * O(x^n)), n))}; /* _Michael Somos_, Sep 17 2004 */
%o A029552 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 / (eta(x + A)^3 * eta(x^4 + A)^2), n));} /* _Michael Somos_, Sep 17 2004 */
%o A029552 (PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, 2*n, prod(i=1, k, (1 -x^(2*n + 1-i)) / (1 - x^i))), n^2-n))}; /* _Michael Somos_, Sep 17 2004 */
%Y A029552 Cf. A083906, A098613.
%K A029552 nonn
%O A029552 0,2
%A A029552 _N. J. A. Sloane_