A244553 - OEIS

%I #13 Sep 08 2022 08:46:08

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

%T -1,4,-2,0,1,0,2,0,-4,2,0,2,2,0,0,0,2,1,-5,4,-4,0,4,0,0,4,-4,2,0,0,0,

%U 0,-1,0,4,2,-2,0,0,0,1,2,-4,2,2,0,0,0,-4,5

%N Expansion of phi(q^2) * (phi(q) - phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.

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

%H G. C. Greubel, <a href="/A244553/b244553.txt">Table of n, a(n) for n = 1..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 * f(-q, -q^7)^2 * phi(q^2) / psi(-q) = q * f(-q, -q^7)^2 * chi(q^2)^2 / chi(-q) in powers of q where phi(), psi(), f() are Ramanujan theta functions.

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

%F Moebius transform is period 8 sequence [ 1, -2, 1, 0, -1, 2, -1, 0, ...].

%F a(2*n) = - A244554(n). a(2*n + 1) = A113411(n). a(8*n + 1) = A112603(n). a(8*n + 3) = 2 * A033761(n). a(8*n + 5) = a(8*n + 7) = 0.

%e G.f. = q - q^2 + 2*q^3 - q^4 + 2*q^6 - q^8 + 3*q^9 - 4*q^10 + 2*q^11 + ...

%t a[ n_] := If[ n < 1, 0, Sum[ {1, -2, 1, 0, -1, 2, -1, 0}[[ Mod[ d, 8, 1] ]], {d, Divisors @ n}]];

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^2] (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^2]) / 2, {q, 0, n}];

%o (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, [0, 1, -2, 1, 0, -1, 2, -1][d%8 + 1]))};

%o (PARI) {a(n) = my(A, B); if( n<0, 0, A = sum(k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n)); B = subst(A, x, x^2); polcoeff( B * (A - B) / 2, n))};

%o (Sage) A = ModularForms( Gamma1(8), 1, prec=33) . basis(); A[1] - A[2];

%o (Magma) A := Basis( ModularForms( Gamma1(8), 1), 33); A[2] - A[3];

%Y Cf. A033761, A113411, A112603, A244554.

%K sign

%O 1,3

%A _Michael Somos_, Jun 30 2014