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A259896 Expansion of psi(x) * psi(x^6) in powers of x where psi() is a Ramanujan theta function.

%I A259896 #17 Feb 16 2025 08:33:26
%S A259896 1,1,0,1,0,0,2,1,0,1,1,0,1,0,0,1,1,0,1,1,0,3,0,0,1,0,0,1,2,0,0,0,0,1,
%T A259896 1,0,2,1,0,2,0,0,2,0,0,1,2,0,0,0,0,2,0,0,1,1,0,1,0,0,1,2,0,2,1,0,2,0,
%U A259896 0,0,1,0,2,1,0,1,0,0,1,0,0,2,0,0,2,0,0
%N A259896 Expansion of psi(x) * psi(x^6) in powers of x where psi() is a Ramanujan theta function.
%C A259896 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A259896 Also the number of positive odd solutions to equation x^2 + 6*y^2 = 8*n + 7. - _Seiichi Manyama_, May 28 2017
%H A259896 Seiichi Manyama, <a href="/A259896/b259896.txt">Table of n, a(n) for n = 0..10000</a>
%H A259896 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A259896 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A259896 Expansion of q^(-7/8) * eta(q^2)^2 * eta(q^12)^2 / (eta(q) * eta(q^6)) in powers of q.
%F A259896 Euler transform of period 12 sequence [ 1, -1, 1, -1, 1, 0, 1, -1, 1, -1, 1, -2, ...].
%F A259896 a(3*n + 1) = A259895(n). a(3*n + 2) = a(9*n + 4) = 0.
%e A259896 G.f. = 1 + x + x^3 + 2*x^6 + x^7 + x^9 + x^10 + x^12 + x^15 + x^16 + ...
%e A259896 G.f. = q^7 + q^15 + q^31 + 2*q^55 + q^63 + q^79 + q^87 + q^103 + q^127 + ...
%t A259896 a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x^3] / (4 q^(7/8)), {x, 0, n}];
%o A259896 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^12 + A)^2 / (eta(x + A) * eta(x^6 + A)), n))};
%Y A259896 Cf. A259895.
%K A259896 nonn
%O A259896 0,7
%A A259896 _Michael Somos_, Jul 07 2015