A182173 Number of inequivalent expressions involving n operands.

2, 10, 94, 1466, 31814, 887650, 30259198, 1218864842, 56644903958, 2983300619410, 175598066553166, 11423394497044154, 813897286250604326, 63030237104398839490, 5271647928235911880222, 473558482553909252128298, 45473767604938843870986422, 4648336478135316689480390770

Offset: 1

Author

David Radcliffe, Apr 16 2012

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

• Jingzhe Tang, Table of n, a(n) for n = 1..300 (first 90 terms from David Radcliffe)
• David Radcliffe, Python script
• Ruud H.G. van Tol, Corrected Python script
• Wikipedia, 24 Game
• Zhujun Zhang, A Chinese web page on exponential generating function
• Zhujun Zhang, A Chinese web page on approximation

Crossrefs

Cf. A140606.

Programs

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)}
for(n=1,20,print1(a(n),", ")) \\ Paul D. Hanna, Aug 12 2018 - After formula by Zhujun Zhang

Formula

From Zhujun Zhang, Aug 11 2018: (Start)

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).

a(n) ~ (n/(e*b))^n * sqrt(b)*c/n where b=0.16142418303980816579438744831086877555003744810690... and c=1.8772213095052105788245813534431275116981368728916.... (End)