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A279270 Expansion of phi(-x) * chi(-x)^2 * f(-x^6)^3 in powers of x where phi(), chi(), f() are Ramanujan theta functions.

%I A279270 #10 Feb 16 2025 08:33:37
%S A279270 1,-4,5,-4,10,-16,12,-8,14,-28,21,-8,30,-40,28,-16,21,-52,34,-20,50,
%T A279270 -56,48,-24,38,-72,44,-28,70,-88,56,-24,43,-100,70,-36,80,-112,84,-32,
%U A279270 62,-104,85,-44,110,-136,56,-56,74,-148,102,-40,130,-144,120,-56,64
%N A279270 Expansion of phi(-x) * chi(-x)^2 * f(-x^6)^3 in powers of x where phi(), chi(), f() are Ramanujan theta functions.
%C A279270 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A279270 G. C. Greubel, <a href="/A279270/b279270.txt">Table of n, a(n) for n = 0..1000</a>
%H A279270 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A279270 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A279270 Expansion of q^(-2/3) * eta(q)^4 * eta(q^6)^3 / eta(q^2)^3 in powers of q.
%F A279270 Euler transform of period 6 sequence [ -4, -1, -4, -1, -4, -4, ...].
%F A279270 3 * a(n) = A260301(3*n + 2).
%e A279270 G.f. = 1 - 4*x + 5*x^2 - 4*x^3 + 10*x^4 - 16*x^5 + 12*x^6 - 8*x^7 + ...
%e A279270 G.f. = q^2 - 4*q^5 + 5*q^8 - 4*q^11 + 10*q^14 - 16*q^17 + 12*q^20 + ...
%t A279270 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] QPochhammer[ x, x^2]^2 QPochhammer[ x^6]^3, {x, 0, n}];
%o A279270 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^6 + A)^3 / eta(x^2 + A)^3, n))};
%Y A279270 Cf. A260301.
%K A279270 sign
%O A279270 0,2
%A A279270 _Michael Somos_, Dec 09 2016