A227398 Expansion of chi(x^3) / chi(x) in powers of x where chi() is a Ramanujan theta function.

1, -1, 1, -1, 1, -2, 2, -3, 3, -3, 4, -5, 6, -7, 8, -9, 10, -12, 14, -16, 18, -20, 23, -26, 30, -34, 38, -42, 47, -53, 60, -67, 74, -82, 91, -102, 114, -126, 139, -153, 169, -187, 207, -228, 250, -274, 301, -331, 364, -399, 436, -476, 520, -569, 622, -679

Offset

0,6

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of f(-x^6) / f(x, x^5) in powers of x where f(,) is Ramanujan's general theta function.

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

Euler transform of period 12 sequence [-1, 1, 0, 0, -1, 0, -1, 0, 0, 1, -1, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A098884.

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

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

G.f.: Product_{k>0} ((1 + x^(6*k - 3)) / (1 + x^(2*k - 1))).

G.f.: 1 / Product_{k>0} ((1 + x^(6*k - 1)) * (1 + x^(6*k - 5))).

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

G.f.: (Sum_{k in Z} (-1)^k * x^(3*k * (3*k-1))) / (Sum_{k in Z} x^(k * (3*k - 2))).

a(n) = (-1)^n * A003105(n). Convolution inverse of A098884.

Example

G.f. = 1 - x + x^2 - x^3 + x^4 - 2*x^5 + 2*x^6 - 3*x^7 + 3*x^8 - 3*x^9 + ...
G.f. = 1/q - q^11 + q^23 - q^35 + q^47 - 2*q^59 + 2*q^71 - 3*q^83 + 3*q^95 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 1 / Product[ 1 - (-x)^k + x^(2 k), {k, n}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[ 1 - x^k + x^(2 k), {k, 1, n, 2}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[ 1 + x^k, {k, 3, n, 6}] / Product[ 1 + x^k, {k, 1, n, 2}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ 1 / (Product[ 1 + x^k, {k, 1, n, 6}] Product[ 1 + x^k, {k, 5, n, 6}]), {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[ 1 + (-x)^k, {k, 1, n, 3}] Product[ 1 + (-x)^k, {k, 2, n, 3}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ -x^3, x^6] / QPochhammer[ -x, x^2], {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^2 / (eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^12 + A)), n))};
(PARI) {a(n) = my(A, m); if( n<0, 0, A = x * O(x^n); m = sqrtint(3*n + 1); polcoeff( eta(x^6 + A) / sum(k= -((m-1)\3), (m+1)\3, x^(k * (3*k - 2)), A), n))};

Crossrefs

Cf. A003105, A098884.

Sequence in context: A302984 A029094 A262950 * A003105 A240856 A081166

Adjacent sequences: A227395 A227396 A227397 * A227399 A227400 A227401

Keyword

sign

Author

Michael Somos, Sep 20 2013

Status

approved