A236360 Numerator of the mean of all parts of all partitions of n.

1, 4, 3, 5, 7, 66, 35, 88, 135, 35, 56, 44, 1313, 63, 220, 48, 1683, 3465, 4655, 1254, 4158, 7348, 28865, 2700, 48950, 10556, 13545, 14872, 132385, 168120, 212102, 89056, 111573, 209270, 520905, 323586, 800569, 988570, 1216215, 35560, 1827903, 744436

Offset: 1

Author

Clark Kimberling, Jan 24 2014

Keywords

• nonn
• frac
• easy

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

Examples

			First eight means:  1, 4/3, 3/2, 5/3, 7/4, 66/35, 35/18, 88/43.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A006128, A066186, A236361.

Programs

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 *)

Formula

M(n) = A066186(n)/A006128(n).