A279371 Expansion of F(q) + 4*F(q^2) + 8*F(q^4) in powers of q where F(q) = q * (f(-q) * f(-q^11))^2.

1, 2, -1, 2, 1, -2, -2, -8, -2, 2, 1, -2, 4, -4, -1, 12, -2, -4, 0, 2, 2, 2, -1, 8, -4, 8, 5, -4, 0, -2, 7, -8, -1, -4, -2, -4, 3, 0, -4, -8, -8, 4, -6, 2, -2, -2, 8, -12, -3, -8, 2, 8, -6, 10, 1, 16, 0, 0, 5, -2, 12, 14, 4, -8, 4, -2, -7, -4, 1, -4, -3, 16, 4

Offset

1,2

Comments

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

Fourier expansion of a multiplicative weight 2 cusp form on Gamma_0(44).

Links

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

a(n) is multiplicative with a(11^e) = 1, a(p^e) = a(p) * a(p^(e-1)) - p * a(p^(e-2)) for p != 11.

a(2*n + 1) = A006571(2*n + 1).

Example

G.f. = q + 2*q^2 - q^3 + 2*q^4 + q^5 - 2*q^6 - 2*q^7 - 8*q^8 - 2*q^9 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^11])^2 + 4 q^2 (QPochhammer[ q^2] QPochhammer[ q^22])^2 + 8 q^4 (QPochhammer[ q^4] QPochhammer[ q^44])^2, {q, 0, n}];

Prog

(PARI) {a(n) = my(A, F); if( n<1, 0, A = x * O(x^n); F = x * (eta(x + A) * eta(x^11 + A))^2; polcoeff( F + 4*subst(F, x, x^2) + 8*subst(F, x, x^4), n))};
(Magma) A := Basis( CuspForms( Gamma0(44), 2), 79); A[1] + 2*A[2] - A[3] + 2*A[4];

Crossrefs

Cf. A006571.

Sequence in context: A089610 A102566 A328771 * A134156 A342156 A324884

Adjacent sequences: A279368 A279369 A279370 * A279372 A279373 A279374

Keyword

sign,mult

Author

Michael Somos, Dec 10 2016

Status

approved