A292526 Expansion of x^1 * (1-x^1) / (1+x^1)^2 - x^4 * (1-x^3) * (1+x^3)^2 + x^9 * (1-x^5) / (1+x^5)^2 - ... in powers of x.

0, 1, -3, 5, -8, 9, -11, 16, -15, 18, -24, 21, -23, 32, -30, 29, -41, 33, -35, 53, -39, 41, -56, 48, -54, 65, -51, 53, -72, 66, -64, 80, -63, 65, -102, 69, -72, 103, -75, 90, -104, 81, -83, 117, -111, 89, -120, 96, -95, 146, -99, 112, -143, 105, -126, 144

Offset: 0

Author

Michael Somos, Sep 18 2017

Keywords

• sign

Comments

In [Andrews and Berndt (2005)] take the derivative of equation (12.4.23) with respect to a, set a=-1, and multiply by q.

Examples

			G.f. = x - 3*x^2 + 5*x^3 - 8*x^4 + 9*x^5 - 11*x^6 + 16*x^7 - 15*x^8 + 18*x^9 + ...
		

References

• G. E. Andrews, B. C. Berndt, Ramanujan's lost notebook, Part I, Springer, New York, 2005, MR2135178 (2005m:11001) See p. 278, Equ. (12.4.23).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500

Crossrefs

Cf. A015128, A292511.

Programs

Maple

seq(coeff(series(add(-(-1)^k*x^(k^2)*(1-x^(2*k-1))/(1+x^(2*k-1))^2,k=1..n), x,n+1),x,n),n=0..60); # Muniru A Asiru, Jul 29 2018

Mathematica

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ -(-1)^k x^k^2 (1 - x^(2 k - 1)) / (1 + x^(2 k - 1))^2, {k, Sqrt@n}], {x, 0, n}]];

PARI

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

Formula

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

G.f. of A292511 is the product of the g.f. of A015128 and this g.f. divided by x.