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A261156 Expansion of chi(q) * chi(-q^9) / (chi(-q) * chi(q^9)) in powers of q where chi() is a Ramanujan theta function.

%I A261156 #18 Feb 16 2025 08:33:26
%S A261156 1,2,2,4,6,8,12,16,22,28,36,48,60,76,96,120,150,184,228,280,340,416,
%T A261156 504,608,732,878,1052,1252,1488,1768,2088,2464,2902,3408,3996,4672,
%U A261156 5460,6364,7400,8600,9972,11544,13344,15400,17752,20424,23472,26944,30876,35346
%N A261156 Expansion of chi(q) * chi(-q^9) / (chi(-q) * chi(q^9)) in powers of q where chi() is a Ramanujan theta function.
%C A261156 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A261156 G. C. Greubel, <a href="/A261156/b261156.txt">Table of n, a(n) for n = 0..1000</a>
%H A261156 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A261156 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A261156 Expansion of eta(q^2)^3 * eta(q^9)^2 * eta(q^36) / (eta(q)^2 * eta(q^4) * eta(q^18)^3) in powers of q.
%F A261156 G.f. A(x) = B(x) / B(x^9) where B(x) is the g.f. of A080054.
%F A261156 Euler transform of period 36 sequence [ 2, -1, 2, 0, 2, -1, 2, 0, 0, -1, 2, 0, 2, -1, 2, 0, 2, 0, 2, 0, 2, -1, 2, 0, 2, -1, 0, 0, 2, -1, 2, 0, 2, -1, 2, 0, ...].
%F A261156 a(n) = 2 * A233693(n) unless n=0.  a(2*n) = 2 * A123629(n) = 2 * A212484(n) unless n=0.
%F A261156 a(3*n) = A186924(n). a(3*n) = 4 * A187100(n) unless n=0.
%F A261156 a(n) = (-1)^n * A260215(n). - _Michael Somos_, Aug 14 2015
%F A261156 a(n) ~ exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - _Vaclav Kotesovec_, Nov 16 2017
%e A261156 G.f. = 1 + 2*x + 2*x^2 + 4*x^3 + 6*x^4 + 8*x^5 + 12*x^6 + 16*x^7 + 22*x^8 + ...
%t A261156 a[ n_] := SeriesCoefficient[ QPochhammer[ -q, q^2] QPochhammer[ -q, q] QPochhammer[ q^9, q^18] QPochhammer[ q^9, -q^9], {q, 0, n}];
%o A261156 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^9 + A)^2 * eta(x^36 + A) / (eta(x + A)^2 * eta(x^4 + A) * eta(x^18 + A)^3), n))};
%Y A261156 Cf. A080054, A123629, A186924, A187100, A212484, A233693, A260215.
%K A261156 nonn
%O A261156 0,2
%A A261156 _Michael Somos_, Aug 10 2015