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A261154 Expansion of psi(q^6) * f(-q^12) / (psi(-q) * psi(q^9)) in powers of q where psi(), f() are Ramanujan theta functions.

%I A261154 #14 Feb 16 2025 08:33:26
%S A261154 1,1,1,2,3,4,6,8,11,14,18,24,30,38,48,60,75,92,114,140,170,208,252,
%T A261154 304,366,439,526,626,744,884,1044,1232,1451,1704,1998,2336,2730,3182,
%U A261154 3700,4300,4986,5772,6672,7700,8876,10212,11736,13472,15438,17673,20207
%N A261154 Expansion of psi(q^6) * f(-q^12) / (psi(-q) * psi(q^9)) in powers of q where psi(), f() are Ramanujan theta functions.
%C A261154 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A261154 Vaclav Kotesovec, <a href="/A261154/b261154.txt">Table of n, a(n) for n = 0..2000</a>
%H A261154 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A261154 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A261154 Expansion of eta(q^2) * eta(q^9) * eta(q^12)^3 / (eta(q) * eta(q^4) * eta(q^6) * eta(q^18)^2) in powers of q.
%F A261154 Euler transform of period 36 sequence [1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, -1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, -1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, ...].
%F A261154 G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 1/2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A186115.
%F A261154 a(n) = A233693(n) unless n=0. a(2*n) = A212484(n).
%F A261154 a(n) ~ exp(2*Pi*sqrt(n)/3) / (4*sqrt(3)*n^(3/4)). - _Vaclav Kotesovec_, Nov 16 2017
%e A261154 G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 11*x^8 + ...
%t A261154 a[ n_] := SeriesCoefficient[ 2^(1/2) q^(1/2) EllipticTheta[ 2, 0, q^3] QPochhammer[ q^12] / (EllipticTheta[ 2, Pi/4, q^(1/2)] EllipticTheta[ 2, 0, q^(9/2)]), {q, 0, n}];
%o A261154 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^9 + A) * eta(x^12 + A)^3 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^18 + A)^2), n))};
%Y A261154 Cf. A186115, A212484, A233693.
%K A261154 nonn
%O A261154 0,4
%A A261154 _Michael Somos_, Aug 10 2015