A255365 Expansion of phi(-x^3) * phi(-x^48) / chi(-x^16) in powers of x where phi(), chi() are Ramanujan theta functions.

1, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0, 1, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 3, 0, 0, -2

Offset: 0

Author

Michael Somos, Feb 21 2015

Keywords

• sign

Comments

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

Examples

			G.f. = 1 - 2*x^3 + 2*x^12 + x^16 - 2*x^19 - 2*x^27 + 2*x^28 + x^32 + ...
G.f. = q^2 - 2*q^11 + 2*q^38 + q^50 - 2*q^59 - 2*q^83 + 2*q^86 + q^98 + ...
		

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

Crossrefs

Cf. A002448, A227395.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^16, x^16] EllipticTheta[ 4, 0, x^3] EllipticTheta[ 3, 0, x^48], {x, 0, n}];
a[n_]:=SeriesCoefficient[EllipticTheta[3,0,-x^3]*EllipticTheta[3,0,-x^48 ]/QPochhammer[x^16, x^32], {x, 0, n}]; Table[a[n], {n,0,100}] (* G. C. Greubel, Mar 14 2018 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^32 + A) * eta(x^48 + A)^2 / (eta(x^6 + A) * eta(x^16 + A) * eta(x^96 + A)), n))};

PARI

{a(n) = my(A, p, e); if( n<0 || n%4==1, 0, n = 3*n+2; A = factor(n); - prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, -(e==1), if( p==3, 0, if( p%8 < 4, e+1, 1-e%2))))))};

Formula

Expansion of q^(-2/3) * eta(q^3)^2 * eta(q^32) * eta(q^48)^2 / (eta(q^6) * eta(q^16) * eta(q^96)) in powers of q.

Euler transform of a period 96 sequence.

a(4*n + 1) = a(4*n + 2) = a(8*n + 7) = a(16*n + 4) = a(16*n + 8) = 0.

a(3*n) = A002448(n). a(3*n + 32) = A227395(n).