A255175 Expansion of phi(-x) / (1 - x)^2 in powers of x where phi() is a Ramanujan theta function.

1, 0, -1, -2, -1, 0, 1, 2, 3, 2, 1, 0, -1, -2, -3, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3

Offset: 0

Author

Michael Somos, Feb 16 2015

Keywords

• sign

Comments

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

Examples

			G.f. = 1 - x^2 - 2*x^3 - x^4 + x^6 + 2*x^7 + 3*x^8 + 2*x^9 + x^10 - x^12 + ...
		

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

Crossrefs

Cf. A053615, A134451, A196199, A329116 (essentially the same), A339265 (first differences).

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] / (1 - x)^2, {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^(Mod[k, 2] + 1), {k, 2, n}], {x, 0, n}];
a[ n_] := If[ n < 0, 0, With[{m = Floor[ Sqrt[ n + 1]]}, (-1)^m (n + 1 - m - m^2)]];
Table[Sum[(-1)^(Floor[Sqrt[i]]), {i,0,n}], {n,0,50}] (* G. C. Greubel, Dec 22 2016 *)

PARI

{a(n) = my(m); if( n<0, 0, m = sqrtint(n + 1); (-1)^m * (n + 1 - m - m^2))};

PARI

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

Python

from math import isqrt
def A255175(n): return ((1+(t:=isqrt(n)))*t-n-1)*(1 if t&1 else -1) # Chai Wah Wu, Aug 04 2022

Formula

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

A053615(n) = abs(A196199(n)) = abs(a(n-1)).

Euler transform of -A134451.

a(n) = Sum_{i=0..n}( (-1)^(floor(sqrt(i))) ). - John M. Campbell, Dec 22 2016