cp's OEIS Frontend

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

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