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A260165 Expansion of f(x, x^2) * f(x, x^3)^3 in powers of x where f(, ) is Ramanujan's general theta function.

%I A260165 #13 Feb 16 2025 08:33:26
%S A260165 1,4,7,10,13,14,18,22,25,28,26,34,37,36,43,38,49,54,56,58,43,64,67,70,
%T A260165 73,62,79,72,90,88,74,98,97,100,90,84,108,112,115,126,98,108,127,130,
%U A260165 140,110,139,142,126,148,133,154,152,160,163,108,169,182,175,180
%N A260165 Expansion of f(x, x^2) * f(x, x^3)^3 in powers of x where f(, ) is Ramanujan's general theta function.
%C A260165 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A260165 G. C. Greubel, <a href="/A260165/b260165.txt">Table of n, a(n) for n = 0..2500</a>
%H A260165 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A260165 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A260165 Expansion of phi(-x^3) * f(-x^2)^5 / phi(-x)^2 in powers of x where phi(), f() are Ramanujan theta functions.
%F A260165 Expansion of q^(-5/12) * eta(q^2)^7 * eta(q^3)^2 / (eta(q)^4 * eta(q^6)) in powers of q.
%F A260165 Euler transform of period 6 sequence [4, -3, 2, -3, 4, -4, ...].
%F A260165 a(n) = A260158(2*n).
%e A260165 G.f. = 1 + 4*x + 7*x^2 + 10*x^3 + 13*x^4 + 14*x^5 + 18*x^6 + 22*x^7 + 25*x^8 + ...
%e A260165 G.f. = q^5 + 4*q^17 + 7*q^29 + 10*q^41 + 13*q^53 + 14*q^65 + 18*q^77 + 22*q^89 + ...
%t A260165 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^3] QPochhammer[ x^2]^5 / EllipticTheta[ 4, 0, x]^2, {x, 0, n}];
%o A260165 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 * eta(x^3 + A)^2 / (eta(x + A)^4 * eta(x^6 + A)), n))};
%Y A260165 Cf. A260158.
%K A260165 nonn
%O A260165 0,2
%A A260165 _Michael Somos_, Nov 09 2015