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A182173 Number of inequivalent expressions involving n operands.

%I A182173 #31 Oct 23 2022 22:55:06
%S A182173 2,10,94,1466,31814,887650,30259198,1218864842,56644903958,
%T A182173 2983300619410,175598066553166,11423394497044154,813897286250604326,
%U A182173 63030237104398839490,5271647928235911880222,473558482553909252128298,45473767604938843870986422,4648336478135316689480390770
%N A182173 Number of inequivalent expressions involving n operands.
%C A182173 Each operand must be used exactly once, and the only allowed operations are addition, subtraction, multiplication, division, and unary minus. Parentheses are permitted. This sequence differs from A140606 by allowing unary minus.
%H A182173 Jingzhe Tang, <a href="/A182173/b182173.txt">Table of n, a(n) for n = 1..300</a> (first 90 terms from David Radcliffe)
%H A182173 David Radcliffe, <a href="/A182173/a182173.py.txt">Python script</a>
%H A182173 Ruud H.G. van Tol, <a href="/A182173/a182173.py3.txt">Corrected Python script</a>
%H A182173 Wikipedia, <a href="http://en.wikipedia.org/wiki/24_Game">24 Game</a>
%H A182173 Zhujun Zhang, <a href="https://zhuanlan.zhihu.com/p/34058293">A Chinese web page on exponential generating function</a>
%H A182173 Zhujun Zhang, <a href="https://zhuanlan.zhihu.com/p/34261468">A Chinese web page on approximation</a>
%F A182173 From _Zhujun Zhang_, Aug 11 2018: (Start)
%F A182173 E.g.f: A(x) = B(x) + C(x) - 2*x, where B(x) = 2*x + exp(C(x)) - 1 - C(x) and C(x) = 2*x + 2*exp(B(x)) - 2*exp(B(x)/2) - B(x).
%F A182173 a(n) ~ (n/(e*b))^n * sqrt(b)*c/n where b=0.16142418303980816579438744831086877555003744810690... and c=1.8772213095052105788245813534431275116981368728916.... (End)
%e A182173 When n=2, there are 10 inequivalent expressions: a+b, a-b, b-a, -a-b, a*b, -a*b, a/b, -a/b, b/a, -b/a.
%o A182173 (PARI) {a(n) = my(A,B=x +x*O(x^n),C=x +x*O(x^n)); for(i=1,n, B = 2*x + exp(C) - 1 - C; C = 2*x + 2*exp(B) - 2*exp(B/2) - B ); A = B + C - 2*x; n!*polcoeff(A,n)}
%o A182173 for(n=1,20,print1(a(n),", ")) \\ _Paul D. Hanna_, Aug 12 2018 - After formula by _Zhujun Zhang_
%Y A182173 Cf. A140606.
%K A182173 nonn
%O A182173 1,1
%A A182173 _David Radcliffe_, Apr 16 2012