A259660 Expansion of f(-x, -x^11) * psi(-x^3)^2 / psi(-x) in powers of x where psi(), f() are Ramanujan theta functions.

1, 0, 0, -1, 1, 1, 0, 0, 1, 0, 0, -1, 1, 0, 0, 1, 1, -1, 0, -1, 2, 1, 0, 0, 0, 1, 0, -1, 1, 0, 0, -1, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, -1, 1, 0, 0, 0, 1, -1, 0, -1, 0, 1, 0, 2, 1, 0, 0, -2, 2, 0, 0, 0, 1, 1, 0, -1, 0, 1, 0, 0, 2, -1, 0, 0, 1, 0, 0, 0, 1, -1

Offset

0,21

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of f(-x, -x^11) * f(x, x^5)^2 / f(x) in powers of x where f(,) is the Ramanujan general theta function.

Euler transform of period 12 sequence [ 0, 0, -1, 1, 1, 0, 1, 1, -1, 0, 0, -2, ...].

a(4*n) = A121444(n). a(4*n + 1) = a(n - 1). a(4*n + 2) = 0.

Convolution of A247133 and A259529.

Example

G.f. = 1 - x^3 + x^4 + x^5 + x^8 - x^11 + x^12 + x^15 + x^16 - x^17 - x^19 + ...
G.f. = q^5 - q^14 + q^17 + q^20 + q^29 - q^38 + q^41 + q^50 + q^53 - q^56 + ...

Mathematica

a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ 0, 2, 0, 0, 1, -1, -1, 0, -1, -1, 1, 0}[[Mod[ k, 12, 1]]], {k, n}], {x, 0, n}];
QP:= QPochhammer; a[n_]:= SeriesCoefficient[(QP[x, x^12]*QP[x^11, x^12]* QP[x^12]*QP[x^3, -x^3]^2*QP[x^6]^2)/(QP[x, -x]*QP[x^2]), {x, 0, n}]; Table[a[n], {n, 0, 100}] (* G. C. Greubel, Mar 17 2018 *)

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k)^([ 2, 0, 0, 1, -1, -1, 0, -1, -1, 1, 0, 0][k%12 + 1]), 1 + x * O(x^n)), n))};

Crossrefs

Cf. A121444, A247133, A259529.

Sequence in context: A288424 A127325 A368750 * A119842 A015624 A015114

Adjacent sequences: A259657 A259658 A259659 * A259661 A259662 A259663

Keyword

sign

Author

Michael Somos, Jul 02 2015

Status

approved