cp's OEIS Frontend

A140606 Number of inequivalent expressions involving n operands.

%I A140606 #41 Oct 23 2022 23:55:06
%S A140606 1,6,68,1170,27142,793002,27914126,1150212810,54326011414,
%T A140606 2894532443154,171800282010062,11243812043430330,804596872359480358,
%U A140606 62506696942427106498,5239819196582605428254,471480120474696200252970,45328694990444455796547766,4637556923393331549190920306
%N A140606 Number of inequivalent expressions involving n operands.
%C A140606 Only binary operators +, -, *, / (and parentheses) are permitted.
%C A140606 The expression (a+b)-c is equivalent to a-c+b and a+b-c but (a+b)-c is different from a-b+c.
%C A140606 Another example: the expression (a-b)/(c-d) is equivalent to (b-a)/(d-c).
%H A140606 Jingzhe Tang, <a href="/A140606/b140606.txt">Table of n, a(n) for n = 1..300</a>
%H A140606 Author?, <a href="http://bbs.emath.ac.cn/viewthread.php?tid=461&amp;page=5&amp;fromuid=20#pid7789">A Chinese web page containing the first 100 terms and C source code to generate it</a>
%H A140606 Author?, <a href="http://tieba.baidu.com/f?kz=239846151">A Chinese web page where the problem originated</a>
%H A140606 Ruud H.G. van Tol, <a href="/A140606/a140606_2.pl.txt">Perl port of the C source code</a>
%H A140606 Zhujun Zhang, <a href="https://zhuanlan.zhihu.com/p/34058293">A Chinese web page on exponential generating function</a>
%H A140606 Zhujun Zhang, <a href="https://zhuanlan.zhihu.com/p/34261468">A Chinese web page on approximation</a>
%F A140606 From _Zhujun Zhang_, Aug 11 2018: (Start)
%F A140606 E.g.f.: A(x) = B(x)+C(x)-D(x)-E(x)-x, where B(x) = 2x+exp(C(x))-1-C(x), C(x) = 2x+2*exp(B(x))-2*exp(B(x)/2)-B(x), D(x) = x+exp(E(x))-1-E(x), and E(x) = x+exp(2*D(x))-exp(D(x))-D(x).
%F A140606 a(n) ~ (n/(e*b))^n * sqrt(b)*c/n, where b=0.16142418303980816579438744831086877555003744810690... and c=1.8772213095052105788245813534431275116981368728916.... (End)
%e A140606 When n=2, there are six inequivalent expressions: a+b, a-b, b-a, a*b, a/b, b/a. Other expressions are equivalent to these (e.g.  b+a is equivalent to a+b).
%Y A140606 Cf. A182173, A351922 (if exponentiation is included).
%K A140606 nonn,nice
%O A140606 1,2
%A A140606 _Zhao Hui Du_, Jul 07 2008