A280822 Expansion of phi(-x) * f(-x)^2 * f(-x^6)^3 in powers of x where phi(), chi(), f() are Ramanujan theta functions.

1, -4, 3, 4, -1, -4, -11, 20, -9, -8, 8, 12, 16, -28, -3, -16, -7, 0, 37, -12, 32, 40, -37, -32, -65, 44, -16, -8, 41, 48, -34, 8, -39, 4, 64, -40, 24, -4, 3, -32, -33, -12, 43, -88, 81, -28, -67, 44, 103, 32, 50, 88, -79, -48, -80, 8, -176, -68, 53, 132, -16

Offset

0,2

Comments

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

Links

Seiichi Manyama, 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

Formula

Expansion of phi(-x)^3 * f(-x^6)^3 / chi(-x)^2 in powers of x where phi(), chi(), f() are Ramanujan theta functions.

Expansion of q^(-5/6) * eta(q)^4 * eta(q^6)^3 / eta(q^2) in powers of q.

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

G.f.: Product_{k>0} (1 - x^k)^4 * (1 - x^(2*k))^2 * (1 + x^(2*k) + x^(4*k))^3.

6 * a(n) = A116418(4*n + 3).

Example

G.f. = 1 - 4*x + 3*x^2 + 4*x^3 - x^4 - 4*x^5 - 11*x^6 + 20*x^7 - 9*x^8 + ...
G.f. = q^5 - 4*q^11 + 3*q^17 + 4*q^23 - q^29 - 4*q^35 - 11*q^41 + 20*q^47 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] QPochhammer[ x]^2 QPochhammer[ x^6]^3, {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x]^3 QPochhammer[ -x, x]^2 QPochhammer[ 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)^4 * eta(x^6 + A)^3 / eta(x^2 + A), n))};
(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k)^4 * (1 - x^(2*k))^2 * (1 + x^(2*k) + x^(4*k))^3, 1 + x * O(x^n)), n))};

Crossrefs

Cf. A116418.

Sequence in context: A195597 A143505 A245727 * A346785 A284517 A286953

Adjacent sequences: A280819 A280820 A280821 * A280823 A280824 A280825

Keyword

sign

Author

Michael Somos, Jan 08 2017

Status

approved