A117586 Coefficients of q in series expansion of Zagier's identity.

0, -1, -2, -1, -1, 2, 0, 4, 1, 2, 1, 2, -4, 1, -1, -5, -2, -1, -3, -1, -2, -2, 5, 0, -1, 1, 8, 0, 3, 2, 2, 2, 3, 0, 4, -7, 0, 0, 2, -3, -8, -2, -1, -3, -2, -4, 0, -3, -3, -2, -1, 7, -1, 0, 1, -1, 0, 12, 2, 2, 0, 4, 3, 4, 0, 2, 4, 3, 0, 5, -12, 2, 0, 1, -1, 1, -3, -11, -1, -2, -6, 2, -4, -3, -3, -4, -2, 1, -5, -3, -3, -2, 11, 2, -2, -3, 2, 0, 0, 3, 12, 1

Offset: 0

Author

Eric W. Weisstein, Mar 29 2006

Keywords

• sign

Examples

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

Links

• Robin Chapman, Franklin's argument proves an identity of Zagier, Electron. J. Combin. 7 #R54 (2000).
• Eric Weisstein's World of Mathematics, Zagier's Identity

Crossrefs

Cf. A046746.

Programs

Mathematica

Flatten[{0, CoefficientList[Series[-Sum[x^(n - 1)*(QPochhammer[x^(n + 1), x]^2/QPochhammer[x^(n), x]), {n, 1, 101}], {x, 0, 100}], x]}] (* Mats Granvik, Jan 05 2015 *)
a[ n_] := SeriesCoefficient[ Sum[ QPochhammer[ x] - QPochhammer[ x, x, k], {k, 0, n}], {x, 0, n}]; (* Michael Somos, Jan 07 2015 *)
a[ n_] := SeriesCoefficient[ -Sum[ QPochhammer[ x^k, x] x^k / (1 - x^k)^2, {k, n}], {x, 0, n}]; (* Michael Somos, Jan 07 2015 *)

Formula

Negative of sequence is convolution of A010815 with A046746. - Michael Somos, Jan 07 2015

a(n) = A067661(n) - A067659(n) [Chapman]. - George Beck, May 06 2017