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A213624 Expansion of psi(x)^2 * psi(x^4) in powers of x where psi() is a Ramanujan theta function.

%I A213624 #12 Feb 16 2025 08:33:17
%S A213624 1,2,1,2,3,2,4,4,2,2,5,4,2,6,3,6,7,2,5,4,5,6,6,2,5,10,3,6,10,4,6,8,3,
%T A213624 8,7,6,7,6,4,6,11,6,9,10,3,6,14,4,8,10,8,10,5,6,4,16,7,4,10,4,13,14,7,
%U A213624 8,8,6,10,12,7,12,15,8,8,10,4,6,17,6,10,10
%N A213624 Expansion of psi(x)^2 * psi(x^4) in powers of x where psi() is a Ramanujan theta function.
%C A213624 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A213624 Dickson writes that Liouville proved several related results about sums of triangular number. In particular, that every nonnegative integer is the sum of t1 + t2 + 4*t3 where t1, t2, t3 are trianglular numbers. - _Michael Somos_, Feb 10 2020
%D A213624  L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 23.
%H A213624 G. C. Greubel, <a href="/A213624/b213624.txt">Table of n, a(n) for n = 0..1000</a>
%H A213624 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A213624 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A213624 Expansion of q^(-3/4) * eta(q^2)^4 * eta(q^8)^2 / (eta(q)^2 * eta(q^4)) in powers of q.
%F A213624 Euler transform of period 8 sequence [2, -2, 2, -1, 2, -2, 2, -3, ...].
%e A213624 G.f. = 1 + 2*x + x^2 + 2*x^3 + 3*x^4 + 2*x^5 + 4*x^6 + 4*x^7 + 2*x^8 + 2*x^9 + ...
%e A213624 G.f. = q^3 + 2*q^7 + q^11 + 2*q^15 + 3*q^19 + 2*q^23 + 4*q^27 + 4*q^31 + 2*q^35 + ...
%t A213624 a[ n_] := SeriesCoefficient[ 1/8 EllipticTheta[ 2, 0, q]^2 EllipticTheta[ 2, 0, q^4], {q, 0, 2 n + 3/2}];
%o A213624 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^8 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)), n))};
%Y A213624 Cf. A008443, A045818,
%K A213624 nonn
%O A213624 0,2
%A A213624 _Michael Somos_, Jun 16 2012