OFFSET
0,2
COMMENTS
M_2(n) is defined to be Sum_{m=-n..n} m^2 M(m,n) where M(m,n) is the number of partitions of n with crank m except for n=1 where M(-1,1) = M(1,1) = -M(0,1) = 1. - Michael Somos, Nov 10 2013
From Omar E. Pol, Jul 25 2022: (Start)
Apart from the initial zero this is also:
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..5000
F. G. Garvan, Higher-order spt functions, arXiv:1008.1207 [math.NT], 2010.
F. G. Garvan, Higher-order spt functions, Adv. Math. 228 (2011), no. 1, 241-265.
Wikipedia, Crank of a partition
FORMULA
a(n) = n*A139582(n). - Omar E. Pol, Jan 03 2013
a(n) ~ exp(Pi*sqrt(2*n/3))/(2*sqrt(3)) * (1 - (sqrt(3/2)/Pi + Pi/(24*sqrt(6))) / sqrt(n)). - Vaclav Kotesovec, Oct 24 2016
EXAMPLE
G.f. = 2*x + 8*x^2 + 18*x^3 + 40*x^4 + 70*x^5 + 132*x^6 + 210*x^7 + ...
For n=1, M_2(1) = Sum_{m=-1..1} m^2 * M(m,2) = (-1)^2*1 + 0^2*(-1) + 1^2*1 = 2. For n=2, the partition [2] has crank 2 and partition [1,1] has crank -2, hence M_2(2) = 2^2 + (-2)^2 = 8. - Michael Somos, Nov 10 2013
MATHEMATICA
a[ n_] := 2 n PartitionsP @ n (* Michael Somos, Nov 10 2013 *)
PROG
(PARI) {a(n) = if( n<0, 0, 2 * n * polcoeff( 1 / eta(x + x * O(x^n)), n))} /* Michael Somos, Nov 10 2013 */
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jan 02 2013
STATUS
approved