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A204770 Expansion of psi(x^3) * f(-x) in powers of x where psi(), f() are Ramanujan theta functions. 2
1, -1, -1, 1, -1, 0, 0, 1, 1, 1, 0, -1, -1, 0, 1, -2, 1, 0, 0, -1, -1, -1, 1, 1, -1, 2, 1, 0, 0, 1, 0, 0, -1, -1, 0, 1, 0, 1, -1, 0, 0, 0, -1, -1, 0, 0, -1, -1, 0, -1, 1, 1, 2, -1, 1, 0, 1, 0, -1, 0, 1, 0, 0, 1, -1, -2, 1, 1, 1, 1, -1, 1, 0, -1, 0, 0, 0, -1, -1, -1, -2, 1, 0, 0, 1, -1, -2, 1, -1, 2, 0, 1, 1, 0, 0, 0, 0, 0, -1, -1, 0, 1, 1, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,16

COMMENTS

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

Conjecture: density of zero terms slowly approaches 1.

The absolute value of the terms grows slowly. The first two with abs(a(n)) = 3 or 4 are a(570) = 3. a(940) = -4.

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 q^(-5/12) * eta(q) * eta(q^6)^2 / eta(q^3) in powers of q.

Euler transform of period 6 sequence [-1, -1, 0, -1, -1, -2, ...].

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

G.f.: Product_{k > 0} (1 - x^k) * (1 + x^(3*k)) * (1 - x^(6*k)). - Michael Somos, May 24 2015

EXAMPLE

G.f. = 1 - x - x^2 + x^3 - x^4 + x^7 + x^8 + x^9 - x^11 - x^12 + x^14 - 2*x^15 + ...

G.f. = q^5 - q^17 - q^29 + q^41 - q^53 + q^89 + q^101 + q^113 - q^137 - q^149 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^6]^2 / QPochhammer[ x^3], {x, 0, n}]; (* Michael Somos, May 24 2015 *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^2 / eta(x^3 + A), n))};

(PARI) {a(n) = my(A, p, e, x, w=quadgen(-8)); if( n<0, 0, n = 12*n + 5; A = factor(n); simplify( 1/w * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, 0, p%12 == 11, !(e%2), p%12 == 5, forstep(y = 3, sqrtint(2*p), 6, if( issquare(2*p - y^2, &x), if( x%6==5, x=-x); x = (x-1)/6; break)); (-1)^(e*x) * [1, w, -1, 0, -1, -w, 1, 0][e%8+1], x=0; forstep(y = 0, sqrtint(p), 6, if( issquare(p - y^2, &x), if( x%6==5, x=-x); x = [x-1, y]/6; break)); if( x==0, (-1)^(e\2) * !(e%2), (-1)^(e*(x[1] + x[2])) * (e+1))))))}; /* Michael Somos, May 24 2015 */

CROSSREFS

Cf. A258196.

Sequence in context: A249223 A115953 A225245 * A333382 A143379 A254605

Adjacent sequences:  A204767 A204768 A204769 * A204771 A204772 A204773

KEYWORD

sign

AUTHOR

Michael Somos, Jan 19 2012

STATUS

approved

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Last modified October 25 07:16 EDT 2021. Contains 348239 sequences. (Running on oeis4.)