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A143434 Expansion of f(x, -x^3) in powers of x where f(,) is Ramanujan's two-variable theta function. 2
1, 1, 0, -1, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

FORMULA

Euler transform of period 16 sequence [ 1, -1, -1, 1, -1, 0, 1, -2, 1, 0, -1, 1, -1, -1, 1, -1, ...].

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

a(n) = (-1)^n * A143433(n).

EXAMPLE

1 + x - x^3 - x^6 - x^10 - x^15 + x^21 + x^28 + x^36 + x^45 - x^55 - x^66 + ...

q + q^9 - q^25 - q^49 - q^81 - q^121 + q^169 + q^225 + q^289 + q^361 + ...

MATHEMATICA

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ (Series[ EllipticTheta[ 3, Log[y] / (2 I), I x^2], {x, 0, n + Floor@Sqrt[n]}] // Normal // TrigToExp) /. {y -> I x}, {x, 0, n}]]

PROG

(PARI) {a(n) = if( n<0, 0, if( issquare( 8*n + 1, &n), n = n\2; (-1)^((n + 2) \ 4), 0))}

(PARI) {a(n) = local(A); if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k)^( [1, -1, 1, 1, -1, 1, 0, -1, 2, -1, 0, 1, -1, 1, 1, -1] [k%16 + 1]), 1 + x * O(x^n)), n))}

CROSSREFS

Cf. A143433.

Sequence in context: A010054 A106459 A143433 * A197870 A033806 A033802

Adjacent sequences:  A143431 A143432 A143433 * A143435 A143436 A143437

KEYWORD

sign

AUTHOR

Michael Somos, Aug 14 2008

STATUS

approved

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Last modified April 10 16:16 EDT 2021. Contains 342845 sequences. (Running on oeis4.)