A280328 Expansion of f(-x)^3 * f(-x^2) * chi(-x^3)^3 in powers of x where chi(), f() are Ramanujan theta functions.

1, -3, -1, 5, 8, 1, -28, -11, 10, 41, 41, -26, -53, -84, 21, 101, 76, -3, -129, -99, 14, 190, 187, -59, -299, -263, 62, 336, 340, -27, -459, -370, 111, 645, 518, -228, -774, -806, 179, 973, 882, -147, -1233, -955, 291, 1565, 1325, -395, -1883, -1767, 338, 2318

Offset: 0

Author

Michael Somos, Dec 31 2016

Keywords

• sign

Comments

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

Examples

			G.f. = 1 - 3*x - x^2 + 5*x^3 + 8*x^4 + x^5 - 28*x^6 - 11*x^7 + 10*x^8 + ...
G.f. = q^-1 - 3*q^5 - q^11 + 5*q^17 + 8*q^23 + q^29 - 28*q^35 - 11*q^41 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• Amanda Clemm, Modular Forms and Weierstrass Mock Modular Forms, Mathematics, volume 4, issue 1, (2016)
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf.A000727, A280384.

Programs

Mathematica

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

PARI

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

Formula

Expansion of q * eta(q^6)^3 * eta(q^12) * eta(q^18)^3 / eta(q^36)^3 in powers of q^6.

Euler transform of period 6 sequence [-3, -4, -6, -4, -3, -4, ...].

a(n) = (-1)^n * A280384(n).

a(5*n + 1) / a(1) == A000727(n) (mod 5). a(125*n + 21) / a(21) == A000727(n) (mod 25).