A261277 Expansion of q^(-1/2) * (eta(q^3)^8 + 4 * eta(q^6)^8)^(1/2) in powers of q.

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

Offset

0,2

Comments

It seems that a( (p - 1)/2 ) = 0 if and only if p is in A167860.

Links

G. C. Greubel, Table of n, a(n) for n = 0..2500

Formula

a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = -2 * (-1)^e if e>0, b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)) if p>3.

G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 72 (t/i)^2 f(t) where q = exp(2 Pi i t).

Convolution square is A261278.

Example

G.f. = 1 + 2*x - 2*x^2 - 2*x^4 + 4*x^5 - 2*x^6 - 4*x^7 + 2*x^8 - 4*x^9 - 8*x^11 + ...
G.f. = q + 2*q^3 - 2*q^5 - 2*q^9 + 4*q^11 - 2*q^13 - 4*q^15 + 2*q^17 - 4*q^19 + ...

Mathematica

a[ n_] := SeriesCoefficient[ Sqrt[ QPochhammer[ x^3]^8 + 4 x QPochhammer[ x^6]^8], {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( sqrt( eta(x^3 + A)^8 + 4 * x * eta(x^6 + A)^8 ), n))};
(Magma) A := Basis( CuspForms( Gamma0(72), 2), 142); A[1] + 2*A[3] - 2*A[4];

Crossrefs

Cf. A167860, A261278.

Sequence in context: A159286 A355837 A372470 * A006462 A278483 A008281

Adjacent sequences: A261274 A261275 A261276 * A261278 A261279 A261280

Keyword

sign

Author

Michael Somos, Aug 14 2015

Status

approved