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%I #31 Oct 23 2022 22:55:06
%S 2,10,94,1466,31814,887650,30259198,1218864842,56644903958,
%T 2983300619410,175598066553166,11423394497044154,813897286250604326,
%U 63030237104398839490,5271647928235911880222,473558482553909252128298,45473767604938843870986422,4648336478135316689480390770
%N Number of inequivalent expressions involving n operands.
%C 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 Jingzhe Tang, <a href="/A182173/b182173.txt">Table of n, a(n) for n = 1..300</a> (first 90 terms from David Radcliffe)
%H David Radcliffe, <a href="/A182173/a182173.py.txt">Python script</a>
%H Ruud H.G. van Tol, <a href="/A182173/a182173.py3.txt">Corrected Python script</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/24_Game">24 Game</a>
%H Zhujun Zhang, <a href="https://zhuanlan.zhihu.com/p/34058293">A Chinese web page on exponential generating function</a>
%H Zhujun Zhang, <a href="https://zhuanlan.zhihu.com/p/34261468">A Chinese web page on approximation</a>
%F From _Zhujun Zhang_, Aug 11 2018: (Start)
%F 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 a(n) ~ (n/(e*b))^n * sqrt(b)*c/n where b=0.16142418303980816579438744831086877555003744810690... and c=1.8772213095052105788245813534431275116981368728916.... (End)
%e 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 (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 for(n=1,20,print1(a(n),", ")) \\ _Paul D. Hanna_, Aug 12 2018 - After formula by _Zhujun Zhang_
%Y Cf. A140606.
%K nonn
%O 1,1
%A _David Radcliffe_, Apr 16 2012
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