A140606 Number of inequivalent expressions involving n operands.

1, 6, 68, 1170, 27142, 793002, 27914126, 1150212810, 54326011414, 2894532443154, 171800282010062, 11243812043430330, 804596872359480358, 62506696942427106498, 5239819196582605428254, 471480120474696200252970, 45328694990444455796547766, 4637556923393331549190920306

Offset: 1

Author

Zhao Hui Du, Jul 07 2008

Keywords

• nonn
• nice

Comments

Only binary operators +, -, *, / (and parentheses) are permitted.

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.

Another example: the expression (a-b)/(c-d) is equivalent to (b-a)/(d-c).

Examples

			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).
		

Links

• Jingzhe Tang, Table of n, a(n) for n = 1..300
• Author?, A Chinese web page containing the first 100 terms and C source code to generate it
• Author?, A Chinese web page where the problem originated
• Ruud H.G. van Tol, Perl port of the C source code
• Zhujun Zhang, A Chinese web page on exponential generating function
• Zhujun Zhang, A Chinese web page on approximation

Crossrefs

Cf. A182173, A351922 (if exponentiation is included).

Formula

From Zhujun Zhang, Aug 11 2018: (Start)

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).

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