OFFSET
1,2
COMMENTS
The arithmetic mean, M(n), of all parts of all partitions of n can be approximated by n^e(n), as typified by these pairs:
n ..... 100 .... 1000 .... 2000 .... 3000 .... 4000 .... 5000
e(n) .. 0.331 .. 0.3410 .. 0.3447 .. 0.3468 .. 0.3483 .. 0.3495
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000
EXAMPLE
First eight means: 1, 4/3, 3/2, 5/3, 7/4, 66/35, 35/18, 88/43.
MAPLE
b:= proc(n, i) option remember; `if`(n=0, [1, 0$2],
`if`(i<1, [0$3], b(n, i-1)+`if`(i>n, [0$3],
(l-> l+[0, l[1]*i, l[1]])(b(n-i, i)))))
end:
a:= n-> numer((l->l[2]/l[3])(b(n$2))):
seq(a(n), n=1..50); # Alois P. Heinz, Feb 06 2014
MATHEMATICA
f[n_] := Sum[DivisorSigma[0, m] PartitionsP[n - m], {m, 1, n}]; u = PartitionsP[Range[50]] Range[50]; t = Table[u[[n]]/f[n], {n, 1, 50}]
Numerator[t] (*A236360*)
Denominator[t] (*A234361*)
means = Map[Mean[Flatten[IntegerPartitions[#]]] &, Range[50]]; pwrLaw = a x^b; fit = FindFit[means, pwrLaw, {a, b}, x]; Show[{ListPlot[means], Plot[Function[{x}, Evaluate[pwrLaw /. fit]][x], {x, 1, Length[means]}]}]
fit (* Peter J. C. Moses, Jan 22 2014 *)
CROSSREFS
KEYWORD
nonn,frac,easy
AUTHOR
Clark Kimberling, Jan 24 2014
STATUS
approved