A306668 - OEIS

%I #16 Apr 02 2021 11:50:04

%S 0,-1,1,0,3,4,20,50,189,588,2100,7116,25344,89298,321178,1156298,

%T 4206059,15356796,56424836,208137800,771229684,2867771004,10700980956,

%U 40050890172,150328400292,565699287186,2133889856550,8067040670100,30559571239890,115986196679730

%N Difference between numbers of binary bracketings of 0^0^...^0 with n 0's giving the result 1 and those giving the result 0, with conventions that 0^0=1^0=1^1=1, 0^1=0.

%C The total number of binary bracketings of 0^0^...^0 with n 0's is A000108(n-1) for n > 0.

%H Alois P. Heinz, <a href="/A306668/b306668.txt">Table of n, a(n) for n = 0..1670</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BinaryBracketing.html">Binary Bracketing</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero">Zero to the power of zero</a>

%F a(n) = A111160(n-1) - A055113(n) for n > 0.

%F a(n) is odd <=> n in { A000079 }.

%e There are A000108(3) = 5 binary bracketings of 0^0^0^0: ((0^0)^0)^0, (0^0)^(0^0), (0^(0^0))^0, 0^((0^0)^0), 0^(0^(0^0)). Only 0^((0^0)^0) evaluates to 0: 0^((0^0)^0) = 0^(1^0) = 0^1 = 0. The four other bracketings evaluate to 1. Thus a(4) = 4-1 = 3.

%p b:= proc(n) option remember; `if`(n<2, [n, 0], add(((f, g)-> [f[1]*g[2],

%p       f[1]*g[1] +f[2]*g[1] +f[2]*g[2]])(b(i), b(n-i)), i=1..n-1))

%p     end:

%p a:= n-> (v-> v[2]-v[1])(b(n)):

%p seq(a(n), n=0..29);

%p # second Maple program:

%p a:= proc(n) option remember; `if`(n<2, -n, ((35*n^3-147*n^2+220*n-120)*

%p       a(n-1)+18*(n-2)*(5*n-6)*(2*n-5)*a(n-2))/((2*(5*n-11))*(2*n-1)*n))

%p     end:

%p seq(a(n), n=0..29);

%t a[n_] := a[n] = If[n<2, -n, ((35n^3 - 147n^2 + 220n - 120) a[n-1] + 18(n-2) (5n - 6)(2n - 5) a[n-2])/((2(5n - 11))(2n - 1)n)];

%t a /@ Range[0, 29] (* _Jean-François Alcover_, Apr 02 2021, after 2nd Maple program *)

%Y Cf. A000079, A000108, A055113, A111160, A211192.

%K sign

%O 0,5

%A _Alois P. Heinz_, Mar 04 2019