A257920 - OEIS

%I #11 Mar 12 2021 22:24:48

%S 1,2,0,1,4,0,0,2,0,3,2,0,2,2,0,0,2,0,3,4,0,0,2,0,0,4,0,2,2,0,1,2,0,0,

%T 6,0,2,0,0,4,0,0,0,2,0,3,4,0,0,4,0,0,2,0,4,2,0,0,2,0,0,2,0,1,4,0,2,6,

%U 0,0,2,0,2,2,0,0,0,0,0,4,0,4,2,0,3,2,0

%N Expansion of phi(x) * psi(x^3) in powers of x where phi(), psi() are Ramanujan theta functions.

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

%H G. C. Greubel, <a href="/A257920/b257920.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^(-3/8) * eta(q^2)^5 * eta(q^6)^2 / (eta(q)^2 * eta(q^3) * eta(q^4)^2) in powers of q.

%F Euler transform of period 12 sequence [ 2, -3, 3, -1, 2, -4, 2, -1, 3, -3, 2, -2, ...].

%F a(n) = A129402(4*n + 1) = A134177(4*n + 1) = A000377(8*n + 3) = A192013(8*n + 3).

%F a(3*n + 2) = 0. a(3*n + 1) = 2 * A128591(n).

%e G.f. = 1 + 2*x + x^3 + 4*x^4 + 2*x^7 + 3*x^9 + 2*x^10 + 2*x^12 + 2*x^13 + ...

%e G.f. = q^3 + 2*q^11 + q^27 + 4*q^35 + 2*q^59 + 3*q^75 + 2*q^83 + 2*q^99 + ...

%t a[ n_] := Seriescoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 2, 0, x^(3/2)] / (2 x^(3/8)), {x, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^6 + A)^2 / (eta(x + A)^2 * eta(x^3 + A) * eta(x^4 + A)^2), n))};

%Y Cf. A000377, A128591, A129402, A134177, A192013.

%K nonn

%O 0,2

%A _Michael Somos_, Jul 12 2015