A280384 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, 1994

Offset

0,2

Comments

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

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

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^12)^10 * eta(q^36)^6 / (eta(q^6)^3 * eta(q^18)^3 * eta(q^24)^3 * eta(q^72)^3) in powers of q^6.

Euler transform of period 12 sequence [3, -7, 6, -4, 3, -10, 3, -4, 6, -7, 3, -4, ...].

a(n) = (-1)^n * A280328(n).

a(5*n + 1) / a(1) == A187076(n) (mod 5). a(125*n + 21) / a(21) == A187076(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^2 + A)^10 * eta(x^6 + A)^6 / (eta(x + A)^3 * eta(x^3 + A)^3 * eta(x^4 + A)^3 * eta(x^12 + A)^3), n))};

Crossrefs

Cf. A187076, A280328.

Sequence in context: A209831 A284367 A280328 * A124420 A176105 A094353

Adjacent sequences: A280381 A280382 A280383 * A280385 A280386 A280387

Keyword

sign

Author

Michael Somos, Jan 01 2017

Status

approved