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A257640 Expansion of psi(x)^2 / phi(-x^3) in powers of x where phi(), psi() are Ramanujan theta functions.

%I A257640 #21 Feb 16 2025 08:33:25
%S A257640 1,2,1,4,6,2,11,14,4,24,30,10,47,58,18,88,108,32,156,188,57,268,318,
%T A257640 94,444,522,152,716,834,244,1129,1308,378,1744,2010,576,2652,3038,870,
%U A257640 3968,4524,1288,5857,6650,1884,8540,9660,2730,12312,13878,3906,17572
%N A257640 Expansion of psi(x)^2 / phi(-x^3) in powers of x where phi(), psi() are Ramanujan theta functions.
%C A257640 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A257640 Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 3, 2nd equation.
%H A257640 G. C. Greubel, <a href="/A257640/b257640.txt">Table of n, a(n) for n = 0..1000</a>
%H A257640 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A257640 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A257640 Expansion of q^(-1/4) * eta(q^2)^4 * eta(q^6) / (eta(q)^2 * eta(q^3)^2) in powers of q.
%F A257640 Euler transform of period 6 sequence [ 2, -2, 4, -2, 2, -1, ...].
%F A257640 a(n) = A053270(n) unless n == 2 (mod 3). a(3*n) = A053270(3*n) - 2 * A256209(n).
%e A257640 G.f. = 1 + 2*x + x^2 + 4*x^3 + 6*x^4 + 2*x^5 + 11*x^6 + 14*x^7 + 4*x^8 + ...
%e A257640 G.f. = q + 2*q^5 + q^9 + 4*q^13 + 6*q^17 + 2*q^21 + 11*q^25 + 14*q^29 + ...
%t A257640 a[ n_] := SeriesCoefficient[ (1/4) x^(-1/4) EllipticTheta[ 2, 0, x^(1/2)]^2 / EllipticTheta[ 4, 0, x^3], {x, 0, n}];
%o A257640 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^6 + A) / (eta(x + A)^2 * eta(x^3 + A)^2), n))};
%o A257640 (PARI) q='q+O('q^99); Vec(eta(q^2)^4*eta(q^6)/(eta(q)^2*eta(q^3)^2)) \\ _Altug Alkan_, Apr 21 2018
%Y A257640 Cf. A053270, A256209.
%K A257640 nonn
%O A257640 0,2
%A A257640 _Michael Somos_, Nov 04 2015