A181102 Coefficients in the q-expansion of the Gamma_0(6) weight -2 meromorphic modular form F(z) (see Formula section for definition).

1, -10, -29, -104, -273, -760, -1685, -4008, -8334, -17560, -34563, -68080, -127210, -238008, -428579, -767808, -1339605, -2322136, -3938840, -6641256, -11004164, -18110800, -29396445, -47399776, -75525219, -119602776, -187488685, -292150064, -451293015, -693184192, -1056544104, -1601892720, -2412131000

Offset

-1,2

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = -1..1000

J. H. Bruinier and K. Ono, Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms [There is an obvious typo in the file-name (brunier instead of bruinier)]

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

F(z) = (1/2)*(E2(z) - 2*E2(2*z) - 3*E2(3*z) + 6*E2(6*z))/( eta(z)*eta(2*z)*eta(3z)*eta(6*z))^2,

where E2(z) = 1 - 24*Sum_{n>=1} sigma(n)*q^n, q = exp(2*Pi*i*z), eta(z) is the usual Dedekind eta-function (see A010815), and i is the imaginary unit.

Expansion of (1/q) * ((phi(-q) * phi(-q^3))^2 - 8 * q * (psi(q) * psi(q^3))^2) / (f(-q) * f(-q^2) * f(-q^3) * f(-q^6))^2 in powers of q where phi(), psi(), f() are Ramanujan theta functions. - Michael Somos, Jul 07 2015

Example

G.f. = 1/q - 10 - 29*q - 104*q^2 - 273*q^3 - 760*q^4 - 1685*q^5 - 4008*q^6 - ...

Maple

with(numtheory); M:=50;
E2:=series(1-24*add(sigma(n)*q^n, n=1..M), q, M);
t1:=series(E2-2*subs(q=q^2, E2)-3*subs(q=q^3, E2)+6*subs(q=q^6, E2), q, M);
eta:=1 : for m from 1 to M do eta:=eta*(1-q^m); od: eta:=series(eta, q, M);
t2:=(eta*subs(q=q^2, eta)*subs(q=q^3, eta)*subs(q=q^6, eta))^2;
t3:=series(t1/(2*q*t2), q, M);

Mathematica

a[ n_] := SeriesCoefficient[ (1/q) ((EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^3])^2 - (EllipticTheta[ 2, 0, q^(1/2)] EllipticTheta[ 2, 0, q^(3/2)])^2/ 2) / (QPochhammer[ q] QPochhammer[ q^2] QPochhammer[ q^3] QPochhammer[ q^6])^2, {q, 0, n}]; (* Michael Somos, Jul 07 2015 *)

Crossrefs

Cf. A030209, A186100.

Sequence in context: A321140 A301571 A027979 * A057456 A002422 A031195

Adjacent sequences: A181099 A181100 A181101 * A181103 A181104 A181105

Keyword

sign

Author

N. J. A. Sloane, Jan 23 2011

Status

approved