OFFSET
0,2
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).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
O-Yeat Chan, Some asymptotics for cranks, Acta Arithmetica 120 (2005), p. 107-143.
Daniel M. Kane, Resolution of a Conjecture of Andrews and Lewis Involving Cranks of Partitions, Proc. Amer. Math. Soc. 132 (2004), 2247-2256.
FORMULA
Expansion of q^(1/24) * eta(q)^2 / eta(q^3) in powers of q. - Michael Somos, Jul 04 2012
G.f.: Product((1-x^n)/(1+x^n+x^(2*n)),n=1..infinity). Euler transform of period 3 sequence [ -2,-2,-1, ...].
a(n) ~ (exp(-Pi*i/9)*exp(-2*Pi*i*n/3) + exp(Pi*i/9)*exp(2*Pi*i*n/3)) * exp(Pi*sqrt(2*n/3)/3) / sqrt(6*n), where i is the imaginary unit [Kane, 2004]. - Vaclav Kotesovec, May 08 2020
EXAMPLE
1 - 2*x - x^2 + 3*x^3 - x^4 + x^5 + 2*x^6 - 3*x^7 - 2*x^8 + 3*x^9 - 3*x^10 + ...
1/q - 2*q^23 - q^47 + 3*q^71 - q^95 + q^119 + 2*q^143 - 3*q^167 - 2*q^191 + ...
MATHEMATICA
nmax = 100; CoefficientList[Series[Product[(1 - x^k)^2/(1 - x^(3*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* or *) nmax = 100; CoefficientList[Series[QPochhammer[x]^2 / QPochhammer[x^3], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 08 2020 *)
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 / eta(x^3 + A), n))} /* Michael Somos, Jul 04 2012 */
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Vladeta Jovovic, Oct 20 2006
STATUS
approved