A261278 Expansion of eta(q^3)^8 + 4 * eta(q^6)^8 in powers of q.

1, 4, 0, -8, 0, 0, 20, -32, 0, 0, 0, 0, -70, 80, 0, 64, 0, 0, 56, 0, 0, 0, 0, 0, -125, -280, 0, -160, 0, 0, 308, 256, 0, 0, 0, 0, 110, 224, 0, 0, 0, 0, -520, 0, 0, 0, 0, 0, 57, -500, 0, 560, 0, 0, 0, -640, 0, 0, 0, 0, 182, 1232, 0, -512, 0, 0, -880, 0, 0, 0, 0

Offset

1,2

Links

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

Formula

a(n) is multiplicative with a(2^(2*k)) = (-8)^k, a(2^(2*k+1)) = 4 * (-8)^k, a(3^e) = 0^e, a(p^(2*k)) = (-p)^(3^k) and a(p^(2*k+1)) = 0 if p == 5 (mod 6), a(p^e) = a(p) * a(p^(e-1)) - p^3 * a(p^(e-2)) if p == 1 (mod 6).

a(3*n) = a(6*n + 5) = 0. a(3*n + 1) = A000731(n). a(4*n) = -8 * a(n). a(6*n + 1) = A153728(n).

Convolution square of A261277.

Example

G.f. = x + 4*x^2 - 8*x^4 + 20*x^7 - 32*x^8 - 70*x^13 + 80*x^14 + ...

Mathematica

a[ n_] := SeriesCoefficient[ x QPochhammer[ x^3]^8 + 4 x^2 QPochhammer[ x^6]^8, {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^3 + A)^8 + 4 * x * eta(x^6 + A)^8, n))};
(Magma) A := Basis( CuspForms( Gamma0(18), 4), 72); A[1] + 4*A[2] - 8*A[4];
(Sage) A = CuspForms( Gamma0(18), 4, prec=20).basis(); A[0] + 4*A[1] - 8*A[3];

Crossrefs

Cf. A000731, A153728, A261277.

Sequence in context: A019111 A182158 A103554 * A340424 A222609 A247848

Adjacent sequences: A261275 A261276 A261277 * A261279 A261280 A261281

Keyword

sign,mult

Author

Michael Somos, Aug 14 2015

Status

approved