OFFSET
1,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000
F. G. Garvan, Higher Order Spt-functions, Adv. Math. 228 (2011), no. 1, 241-265. See Eq. (1.1). - From N. J. A. Sloane, Jan 02 2013
FORMULA
a(n) = N_2(n)/2 (where N_2(n) is the second rank moment, see relation (1.1) of the Garvan reference).
a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)) * (1 - (3*sqrt(6)/(2*Pi) + Pi/(24 * sqrt(6))) / sqrt(n)). - Vaclav Kotesovec, Jul 31 2017
MAPLE
b:= proc(n, i) option remember; `if`(n=0 or i<1, 0,
`if`(irem(n, i, 'r')=0, r, 0)+add(b(n-i*j, i-1), j=0..n/i))
end:
a:= n-> n*combinat[numbpart](n)- b(n, n):
seq(a(n), n=1..60); # Alois P. Heinz, Jan 09 2013
MATHEMATICA
terms = 41; gf = Sum[x^n/(1 - x^n)*Product[1/(1 - x^k), {k, n, terms}], {n, 1, terms}]; spt = CoefficientList[ Series[gf, {x, 0, terms}], x] // Rest; a[n_] := n*PartitionsP[n] - spt[[n]]; Table[a[n], {n, 1, terms}] (* Jean-François Alcover, Jan 17 2013, after g.f. of spt(n) *)
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jan 02 2013
STATUS
approved