A143431 -id:A143431 - cp's OEIS frontend

A143433 Expansion of f(-x, x^3) in powers of x where f(,) is Ramanujan's general theta function.

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

Offset: 0

Michael Somos, Aug 14 2008

sign

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

			G.f. = 1 - x + x^3 - x^6 - x^10 + x^15 - x^21 + x^28 + x^36 - x^45 + x^55 + ...
G.f. = q - q^9 + q^25 - q^49 - q^81 + q^121 - q^169 + q^225 + q^289 + ...

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

Cf. A143434, A244465, A244525.

a[ n_] := SeriesCoefficient[ QPochhammer[ x, -x^4] QPochhammer[ -x^3, -x^4] QPochhammer[ -x^4], {x, 0, n}]; (* Michael Somos, Jun 03 2015 *)

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

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

Euler transform of period 16 sequence [ -1, 0, 1, 1, 1, -1, -1, -2, -1, -1, 1, 1, 1, 0, -1, -1, ...].
Pattern of signs of nonzero terms is A143431.
G.f.: Sum_{k>=0} (-1)^(k + floor(k/4)) * x^(k * (k+1) / 2).
a(n) = (-1)^n * A143434(n).
a(2*n) = A244465(n). a(2*n + 1) = - A244525(n). a(3*n + 2) = a(5*n + 2) = a(5*n + 4) = 0.

A143432 Ultimately period 4 sequence [ 2, 2, 0, 0, ...] with a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 2

Offset: 0

Michael Somos, Aug 14 2008, Sep 18 2008

Index entries for linear recurrences with constant coefficients, signature (1,-1,1)

Convolution inverse of A143431.

PadRight[{1,1},120,{2,2,0,0}] (* Harvey P. Dale, Feb 13 2016 *)

{a(n) = if(n<0, 0, n = n\2; if( n%2, 0, (n>1)+1 ))}

Euler transform of length 8 sequence [ 1, -1, 0, 2, 0, 0, 0, -1].
a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = a(4*n + 1) = 2 unless n < 1.
G.f.: (1 + x^4) / ((1 - x) * (1 + x^2)).
a(n) = ((n+2) mod 4) - (n mod 2) - floor(3/(n+2)). - Wesley Ivan Hurt, Jun 30 2013

A162285 Periodic length 8 sequence [1, -1, -1, 1, -1, 1, 1, -1, ...].

Original entry on oeis.org

1, -1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 1, 1, -1, 1

Offset: 0

Michael Somos, Jun 29 2009

			G.f. = 1 - x - x^2 + x^3 - x^4 + x^5 + x^6 - x^7 + x^8 - x^9 - x^10 + x^11 + ...

Index entries for linear recurrences with constant coefficients, signature (0,0,0,-1).

Cf. A143431.

m:=100; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x)*(1-x^2)/(1+x^4))); // G. C. Greubel, Sep 21 2018

CoefficientList[Series[(1-x)*(1-x^2)/(1+x^4), {x, 0, 100}], x] (* G. C. Greubel, Sep 21 2018 *)
PadRight[{},120,{1,-1,-1,1,-1,1,1,-1}] (* or *) LinearRecurrence[{0,0,0,-1},{1,-1,-1,1},120] (* Harvey P. Dale, May 27 2023 *)

PARI
```
a(n) = (-1)^(n + (n+2)\4)
```

Euler transform of length 8 sequence [ -1, -1, 0, -1, 0, 0, 0, 1].
a(3 - n) = a(n). a(n + 4) = - a(n).
G.f.: (1 - x) * (1 - x^2) / (1 + x^4).
G.f.: 1 / (1 + x / (1 - 2*x / (1 + x / (1 + x / (1 + x^2 / (1 - x)))))). - Michael Somos, May 12 2012
G.f. A(x) = 1 - x / (1 - x / (1 + 2*x / (1 - x^3 / (1 - x / (1 + x / (1 - x)))))). - Michael Somos, Jan 03 2013
a(n) = A143431(n + 2).

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A143433 Expansion of f(-x, x^3) in powers of x where f(,) is Ramanujan's general theta function.

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A143432 Ultimately period 4 sequence [ 2, 2, 0, 0, ...] with a(0) = a(1) = 1.

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A162285 Periodic length 8 sequence [1, -1, -1, 1, -1, 1, 1, -1, ...].

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