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%I #11 Feb 06 2014 09:14:36
%S 1,4,3,5,7,66,35,88,135,35,56,44,1313,63,220,48,1683,3465,4655,1254,
%T 4158,7348,28865,2700,48950,10556,13545,14872,132385,168120,212102,
%U 89056,111573,209270,520905,323586,800569,988570,1216215,35560,1827903,744436
%N Numerator of the mean of all parts of all partitions of n.
%C 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:
%C n ..... 100 .... 1000 .... 2000 .... 3000 .... 4000 .... 5000
%C e(n) .. 0.331 .. 0.3410 .. 0.3447 .. 0.3468 .. 0.3483 .. 0.3495
%H Alois P. Heinz, <a href="/A236360/b236360.txt">Table of n, a(n) for n = 1..1000</a>
%F M(n) = A066186(n)/A006128(n).
%e First eight means: 1, 4/3, 3/2, 5/3, 7/4, 66/35, 35/18, 88/43.
%p b:= proc(n, i) option remember; `if`(n=0, [1, 0$2],
%p `if`(i<1, [0$3], b(n, i-1)+`if`(i>n, [0$3],
%p (l-> l+[0, l[1]*i, l[1]])(b(n-i, i)))))
%p end:
%p a:= n-> numer((l->l[2]/l[3])(b(n$2))):
%p seq(a(n), n=1..50); # _Alois P. Heinz_, Feb 06 2014
%t 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}]
%t Numerator[t] (*A236360*)
%t Denominator[t] (*A234361*)
%t 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]}]}]
%t fit (* _Peter J. C. Moses_, Jan 22 2014 *)
%Y Cf. A006128, A066186, A236361.
%K nonn,frac,easy
%O 1,2
%A _Clark Kimberling_, Jan 24 2014
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