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A236360 Numerator of the mean of all parts of all partitions of n. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A035427 A257120 A033546 * A010475 A256367 A242910

Adjacent sequences:  A236357 A236358 A236359 * A236361 A236362 A236363

KEYWORD

nonn,frac,easy

AUTHOR

Clark Kimberling, Jan 24 2014

STATUS

approved

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Last modified August 17 06:10 EDT 2017. Contains 290635 sequences.