A182056 Expansion of psi(x) * chi(-x^3) * f(-x^16) * chi(-x^24) / phi(-x^12)^2 in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.

1, 1, 0, 0, -1, 0, 0, 0, 0, -2, 0, 0, 4, 4, 0, 0, -6, -1, 0, 0, 1, -8, 0, 0, 11, 14, 0, 0, -19, -4, 0, 0, 4, -23, 0, 0, 31, 40, 0, 0, -50, -10, 0, 0, 11, -60, 0, 0, 77, 98, 0, 0, -122, -24, 0, 0, 28, -140, 0, 0, 173, 224, 0, 0, -273, -54, 0, 0, 62, -304, 0, 0

Offset: 0

Author

Michael Somos, Apr 08 2012

Keywords

• sign

Comments

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

Examples

			1 + x - x^4 - 2*x^9 + 4*x^12 + 4*x^13 - 6*x^16 - x^17 + x^20 + ...
1/q + q^2 - q^11 - 2*q^26 + 4*q^35 + 4*q^38 - 6*q^47 - q^50 + q^59 + ...
		

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. A182032.

Programs

Mathematica

QP := QPochhammer; A182056[n_] := SeriesCoefficient[QP[q^2]^2*QP[q^3]* QP[q^16]*QP[q^24]^3/(QP[q]* QP[q^6]*QP[q^12]^4*QP[q^48]), {q, 0, n}];
Table[A182056[n], {n, 0, 50}] (* G. C. Greubel, Dec 24 2017 *)

PARI

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^16 + A) * eta(x^24 + A)^3 / (eta(x + A) * eta(x^6 + A) * eta(x^12 + A)^4 * eta(x^48 + A)), n))}

Formula

Expansion of q^(-1/3) * eta(q^2)^2 * eta(q^3) * eta(q^16) * eta(q^24)^3 / (eta(q) * eta(q^6) * eta(q^12)^4 * eta(q^48)) in powers of q.

Euler transform of period 48 sequence [ 1, -1, 0, -1, 1, -1, 1, -1, 0, -1, 1, 3, 1, -1, 0, -2, 1, -1, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, -1, 1, -2, 0, -1, 1, 3, 1, -1, 0, -1, 1, -1, 1, -1, 0, -1, 1, 0, ...].

a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A182032(12*n - 1). a(4*n + 1) = A182032(12*n + 2).