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A262151 Expansion of f(-x^3)^3 / (f(x)^2 * f(-x^2)) in powers of x where f() is a Ramanujan theta function.

%I A262151 #14 Feb 16 2025 08:33:27
%S A262151 1,-2,6,-15,33,-68,134,-253,460,-811,1394,-2344,3863,-6253,9964,
%T A262151 -15653,24269,-37178,56331,-84489,125529,-184867,270027,-391391,
%U A262151 563205,-804925,1142998,-1613195,2263675,-3159023,4385502,-6057865,8328200,-11397371,15529768
%N A262151 Expansion of f(-x^3)^3 / (f(x)^2 * f(-x^2)) in powers of x where f() is a Ramanujan theta function.
%C A262151 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A262151 G. C. Greubel, <a href="/A262151/b262151.txt">Table of n, a(n) for n = 0..1000</a>
%H A262151 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A262151 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A262151 Expansion of q^(-5/24) * eta(q)^2 * eta(q^3)^3 * eta(q^4)^2 / eta(q^2)^7 in power of q.
%F A262151 Euler transform of period 12 sequence [ -2, 5, -5, 3, -2, 2, -2, 3, -5, 5, -2, 0, ...].
%F A262151 a(n) ~ (-1)^n * exp(Pi*sqrt(3*n/2)) / (2^(11/4) * 3^(5/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 23 2015
%e A262151 G.f. = 1 - 2*x + 6*x^2 - 15*x^3 + 33*x^4 - 68*x^5 + 134*x^6 - 253*x^7 + ...
%e A262151 G.f. = q^5 - 2*q^29 + 6*q^53 - 15*q^77 + 33*q^101 - 68*q^125 + 134*q^149 + ...
%t A262151 a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^3 / (QPochhammer[ -x]^2 QPochhammer[ x^2]), {x, 0, n}];
%t A262151 nmax = 40; CoefficientList[Series[Product[(1-x^(3*k))^3 * (1+x^(2*k))^2 / ((1-x^k)^3 * (1+x^k)^5), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 13 2015 *)
%o A262151 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^3 + A)^3 * eta(x^4 + A)^2 / eta(x^2 + A)^7, n))};
%K A262151 sign
%O A262151 0,2
%A A262151 _Michael Somos_, Sep 13 2015