A226075 Expansion of (eta(q) * eta(q^11))^2 + 2 * (eta(q^2) * eta(q^22))^2 in powers of q.

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

Offset: 1

Author

Michael Somos, May 25 2013

Keywords

• sign
• mult

Examples

			G.f. = q - q^3 - 2*q^4 + q^5 - 2*q^7 + 4*q^8 - 2*q^9 + q^11 + 2*q^12 + 4*q^13 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..2500

Crossrefs

Cf. A006571, A090132, A214733.

Programs

Magma

Basis( CuspForms( Gamma0(22), 2), 79)[1];

Mathematica

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^11])^2 + 2 q^2 ( QPochhammer[ q^2] QPochhammer[ q^22])^2, {q, 0, n}]; (* Michael Somos, Apr 25 2015 *)

PARI

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^11 + A))^2 + 2 * x * (eta(x^2 + A) * eta(x^22 + A))^2, n))};

Sage

CuspForms( Gamma0(22), 2, prec=79).0;

Formula

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

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

a(4*n + 2) = 0. a(4*n) = -2 * A006571(n). a(2^n) = A090132(n). a(3^n) = A214733(n+1).