A096661 Fine's numbers J(n).

0, 0, -1, 1, -1, 1, -1, 2, -1, 0, -1, 2, -1, 0, -1, 1, -1, 0, 0, 2, -1, -1, -1, 2, 0, 0, 0, 1, -1, 0, -1, 2, -1, -1, 0, 2, 0, 0, -2, 1, -2, 0, 1, 2, -1, 0, -2, 2, 0, 0, -1, 1, -1, 0, -1, 3, -1, 0, 0, 2, -1, 0, -2, 0, -1, 1, 1, 2, -1, 0, -3, 2, 0, 0, 0, 1, -1, -1, -1, 2, -2, 0, 0, 2, 1, 1, -2, 0, -1, 0, 0, 1, -1, 0, -2, 3, 0, 0, 1, 0, -1, 0, -1, 2, -1

Offset: 0

Author

N. J. A. Sloane, Sep 15 2004

Keywords

• sign

Examples

			G.f. = - x^2 + x^3 - x^4 + x^5 - x^6 + 2*x^7 - x^8 - x^10 + 2*x^11 - x^12 + ...
		

References

• N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 62, Eq. (27.1).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• L. A. Dragonette, Some Asymptotic Formulae for the Mock Theta Series of Ramanujan, Trans. Amer. Math. Soc., 72 (1952), 474-500. See page 496.

Crossrefs

Cf. A064053, A097042.

Programs

Maple

add( (-1)^n*q^((3*n^2+n)/2)/(1+q^n),n=1..10);

Mathematica

a[n_]:= SeriesCoefficient[Sum[(-1)^k*q^((3*k^2 + k)/2)/(1 + q^k), {k, 1, 2*nmax}], {q,0,n}]; Table[a[n], {n,0,50}] (* G. C. Greubel, Feb 18 2018 *)

PARI

{a(n) = if( n<0, 0, polcoeff( sum(k=1, (sqrtint(24*n + 1) - 1) \ 6, (-1)^k * x^((3*k^2 + k)/2) / (1 + x^k), x * O(x^n)), n))}; /* Michael Somos, Mar 13 2006 */

Formula

G.f.: Sum_{n >= 1} (-1)^n*q^((3*n^2+n)/2)/(1+q^n).

Dragonette's gamma(n) = A064053(n) = 4*a(n) if n>0.