A124226 Number of partitions of n with even crank minus number of partitions of n with odd crank.

1, -1, 2, -1, 5, -5, 3, -5, 6, -10, 10, -8, 13, -15, 15, -16, 23, -27, 25, -30, 35, -40, 42, -45, 55, -66, 68, -70, 86, -95, 100, -110, 125, -141, 150, -161, 185, -207, 215, -235, 266, -293, 310, -335, 375, -410, 438, -470, 521, -575, 610, -653, 725, -785, 835, -900, 983, -1070, 1140, -1220, 1331, -1443, 1532

Offset: 0

Author

Vladeta Jovovic, Oct 20 2006

Keywords

• easy
• sign

Comments

For a partition p, let l(p) = largest part of p, w(p) = number of 1's in p, m(p) = number of parts of p larger than w(p). The crank of p is given by l(p) if w(p) = 0, otherwise m(p)-w(p).

Examples

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

Crossrefs

Cf. A124227, A124228, A132970.

Programs

Maple

p:=2*q + product((1-q^i)/(1+q^i)^2, i=1..200): s:=series(p, q, 200): for j from 0 to 199 do printf(`%d,`,coeff(s,q, j)) od: # James Sellers, Nov 30 2006

Formula

G.f.: 2*x + Product_{i>=1} (1-x^i)/(1+x^i)^2.

a(n) = A132970(n) unless n=1. - Michael Somos, Jul 27 2015

a(n) ~ (-1)^n * exp(Pi*sqrt(n/6)) / (2*sqrt(n)). - Vaclav Kotesovec, Oct 14 2017

Extensions

More terms from James Sellers, Nov 30 2006