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

1,2

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

Formula

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

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

Cf. A006128, A066186, A236361.

Sequence in context: A257120 A306835 A033546 * A299323 A010475 A256367

Adjacent sequences: A236357 A236358 A236359 * A236361 A236362 A236363

Keyword

nonn,frac,easy

Author

Clark Kimberling, Jan 24 2014

Status

approved