A258832 Expansion of psi(-x^3) * f(-x, x^2) in powers of x where psi(), f(,) are Ramanujan theta functions.

1, -1, 1, -1, 1, -2, 0, -1, 1, -1, 2, -1, 1, 0, 1, -2, 1, 0, 2, -1, 1, -1, 1, -1, 1, -2, 1, 0, 0, -1, 2, -2, 1, -1, 0, -3, 0, -1, 1, 0, 2, 0, 1, -1, 2, -2, 1, -1, 0, -1, 1, -1, 2, -1, 1, 0, 1, -2, 1, 0, 3, 0, 0, -1, 1, -2, 1, -1, 1, -1, 3, -1, 0, -1, 0, -2, 0

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..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-5/12) * eta(q) * eta(q^4) * eta(q^6)^4 / (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, -2, -1, 0, 0, 1, -1, -2, ...].

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

a(n) = (-1)^n * A121444(n). Convolution square is A258831.

Example

G.f. = 1 - x + x^2 - x^3 + x^4 - 2*x^5 - x^7 + x^8 - x^9 + 2*x^10 + ...
G.f. = q^5 - q^17 + q^29 - q^41 + q^53 - 2*q^65 - q^89 + q^101 - q^113 + ...

Mathematica

a[ n_] := If[ n < 0, 0, (-1)^n DivisorSum[ 12 n + 5, KroneckerSymbol[ -4, #] &] / 2];
a[ n_] := SeriesCoefficient[ QPochhammer[ x^6]^2 QPochhammer[ x, -x] / QPochhammer[ x^3, -x^3], {x, 0, n}];

Prog

(PARI) {a(n) = if( n<0, 0, (-1)^n * sumdiv( 12*n + 5, d, kronecker( -4, d)) / 2)};
(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)^4 / (eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^12 + A)), n))};

Crossrefs

Cf. A121444, A258831.

Sequence in context: A281491 A099494 A030341 * A121444 A118230 A179181

Adjacent sequences: A258829 A258830 A258831 * A258833 A258834 A258835

Keyword

sign

Author

Michael Somos, Jun 11 2015

Status

approved