cp's OEIS Frontend

A306668 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.

%I A306668 #17 Feb 16 2025 08:33:55
%S A306668 0,-1,1,0,3,4,20,50,189,588,2100,7116,25344,89298,321178,1156298,
%T A306668 4206059,15356796,56424836,208137800,771229684,2867771004,10700980956,
%U A306668 40050890172,150328400292,565699287186,2133889856550,8067040670100,30559571239890,115986196679730
%N A306668 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 A306668 The total number of binary bracketings of 0^0^...^0 with n 0's is A000108(n-1) for n > 0.
%H A306668 Alois P. Heinz, <a href="/A306668/b306668.txt">Table of n, a(n) for n = 0..1670</a>
%H A306668 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BinaryBracketing.html">Binary Bracketing</a>
%H A306668 Wikipedia, <a href="https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero">Zero to the power of zero</a>
%F A306668 a(n) = A111160(n-1) - A055113(n) for n > 0.
%F A306668 a(n) is odd <=> n in { A000079 }.
%e A306668 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 A306668 b:= proc(n) option remember; `if`(n<2, [n, 0], add(((f, g)-> [f[1]*g[2],
%p A306668       f[1]*g[1] +f[2]*g[1] +f[2]*g[2]])(b(i), b(n-i)), i=1..n-1))
%p A306668     end:
%p A306668 a:= n-> (v-> v[2]-v[1])(b(n)):
%p A306668 seq(a(n), n=0..29);
%p A306668 # second Maple program:
%p A306668 a:= proc(n) option remember; `if`(n<2, -n, ((35*n^3-147*n^2+220*n-120)*
%p A306668       a(n-1)+18*(n-2)*(5*n-6)*(2*n-5)*a(n-2))/((2*(5*n-11))*(2*n-1)*n))
%p A306668     end:
%p A306668 seq(a(n), n=0..29);
%t A306668 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 A306668 a /@ Range[0, 29] (* _Jean-François Alcover_, Apr 02 2021, after 2nd Maple program *)
%Y A306668 Cf. A000079, A000108, A055113, A111160, A211192.
%K A306668 sign
%O A306668 0,5
%A A306668 _Alois P. Heinz_, Mar 04 2019