login
Expansion of psi(x)^2 * psi(x^4) in powers of x where psi() is a Ramanujan theta function.
10

%I #11 Mar 12 2021 22:24:46

%S 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 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 8,8,6,10,12,7,12,15,8,8,10,4,6,17,6,10,10

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

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%C 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 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 G. C. Greubel, <a href="/A213624/b213624.txt">Table of n, a(n) for n = 0..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of q^(-3/4) * eta(q^2)^4 * eta(q^8)^2 / (eta(q)^2 * eta(q^4)) in powers of q.

%F Euler transform of period 8 sequence [2, -2, 2, -1, 2, -2, 2, -3, ...].

%e 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 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 a[ n_] := SeriesCoefficient[ 1/8 EllipticTheta[ 2, 0, q]^2 EllipticTheta[ 2, 0, q^4], {q, 0, 2 n + 3/2}];

%o (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 Cf. A008443, A045818,

%K nonn

%O 0,2

%A _Michael Somos_, Jun 16 2012