A275578 Expansion of (F(x) + F(-x)) / 2 in powers of x^2 where F(x) = (f(-x) * f(-x^11))^2 and f() is a Ramanujan theta function.

1, -1, 1, -2, -2, 1, 4, -1, -2, 0, 2, -1, -4, 5, 0, 7, -1, -2, 3, -4, -8, -6, -2, 8, -3, 2, -6, 1, 0, 5, 12, 4, 4, -7, 1, -3, 4, 4, -2, -10, 1, -6, -2, 0, 15, -8, -7, 0, -7, -2, 2, -16, 2, 18, 10, -3, 9, -1, -8, 4, 1, 8, -9, 8, 6, -18, 0, 5, -7, 10, -8, 4, 0

Offset

0,4

Comments

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

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

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of F(x) + 2*x*F(x^2) + 2*x^3*F(x^4) in powers of x^2 where F(x) = (f(-x) * f(-x^11))^2 and f() is a Ramanujan theta function.

G.f. is a period 1 Fourier series which satisfies f(-1 / (44 t)) = 22 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A279371.

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

Example

G.f. = 1 - x + x^2 - 2*x^3 - 2*x^4 + x^5 + 4*x^6 - x^7 - 2*x^8 + 2*x^10 + ...
G.f. = q - q^3 + q^5 - 2*q^7 - 2*q^9 + q^11 + 4*q^13 - q^15 - 2*q^17 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^11])^2, {x, 0, 2 n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, n = 2*n; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^11 + A))^2, n))};
(Magma) A := Basis( CuspForms( Gamma0(44), 2), 146); A[1] - A[3];
(Sage) A = CuspForms( Gamma0(44), 2, prec=146) . basis(); A[0] - A[2];

Crossrefs

Cf. A006571, A279371.

Sequence in context: A320500 A140957 A197250 * A112085 A353843 A090002

Adjacent sequences: A275575 A275576 A275577 * A275579 A275580 A275581

Keyword

sign

Author

Michael Somos, Dec 25 2016

Status

approved