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

0,2

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..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

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).

Example

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 + ...

Mathematica

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

Prog

(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))};

Crossrefs

Cf.A000727, A280384.

Sequence in context: A063858 A209831 A284367 * A280384 A124420 A176105

Adjacent sequences: A280325 A280326 A280327 * A280329 A280330 A280331

Keyword

sign

Author

Michael Somos, Dec 31 2016

Status

approved