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A255170 a(n) = A087803(n) - n + 1.

%I A255170 #41 Feb 16 2025 08:33:25
%S A255170 1,1,2,5,13,32,79,193,478,1196,3037,7802,20287,53259,141069,376449,
%T A255170 1011295,2732453,7421128,20247355,55469186,152524366,420807220,
%U A255170 1164532203,3231706847,8991343356,25075077684,70082143952,196268698259,550695545855,1547867058852
%N A255170 a(n) = A087803(n) - n + 1.
%C A255170 Conjectured extension of A199812: number of distinct values taken by w^w^...^w (with n w's and parentheses inserted in all possible ways) where w is the first transfinite ordinal omega. So far all known terms of A199812 (that is, 20 of them) coincide with this sequence. It is conjectured that A199812 is actually identical to this sequence, but it remains unproved, and is computationally difficult to check for n > 20.
%H A255170 Alois P. Heinz, <a href="/A255170/b255170.txt">Table of n, a(n) for n = 1..1000</a>
%H A255170 Libor Behounek, <a href="http://mujweb.cz/behounek/logic/papers/ordcalc/index.html">Ordinal Calculator</a>
%H A255170 R. K. Guy and J. L. Selfridge, <a href="/A003018/a003018.pdf">The nesting and roosting habits of the laddered parenthesis</a>
%H A255170 MathOverflow, <a href="http://mathoverflow.net/q/103411/9550">A discussion related to this sequence</a>
%H A255170 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OrdinalNumber.html">Ordinal Number</a>.
%H A255170 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RootedTree.html">Rooted Tree</a>.
%F A255170 a(n) = 1 - n + Sum_{k=1..n} A000081(k).
%F A255170 Recurrence: a(1) = 1, a(n+1) = a(n) + A000081(n+1) - 1.
%F A255170 Recurrence: a(1) = a(2) = 1, a(n) = A174145(n-1) + 2*a(n-1) - a(n-2).
%F A255170 Asymptotics: a(n) ~ c * d^n / n^(3/2), where c = A187770 / (1 - 1 / A051491) = 0.664861... and d = A051491 = 2.955765...
%e A255170 a(4) = 1 - 4 + Sum_{k=1..4} A000081(k) = 1 - 4 + 1 + 1 + 2 + 4 = 5.
%e A255170 a(5) = 1 - 5 + Sum_{k=1..5} A000081(k) = 1 - 5 + 1 + 1 + 2 + 4 + 9 = 13.
%p A255170 with(numtheory):
%p A255170 t:= proc(n) option remember; `if`(n<2, n, (add(add(
%p A255170       d*t(d), d=divisors(j))*t(n-j), j=1..n-1))/(n-1))
%p A255170     end:
%p A255170 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<2, 0,
%p A255170       add(b(n-i*j, i-1)*binomial(t(i)+j-1, j), j=0..n/i)))
%p A255170     end:
%p A255170 a:= proc(n) option remember; `if`(n<3, 1,
%p A255170       b(n-1$2) +2*a(n-1) -a(n-2))
%p A255170     end:
%p A255170 seq(a(n), n=1..40);  # _Alois P. Heinz_, Feb 17 2015
%t A255170 t[1] = a[1] = 1; t[n_] := t[n] = Sum[k t[k] t[n - k m]/(n-1), {k, n}, {m, (n-1)/k}]; a[n_] := a[n] = a[n-1] + t[n] - 1; Table[a[n], {n, 40}] (* _Vladimir Reshetnikov_, Aug 12 2016 *)
%Y A255170 Cf. A199812 (conjectured to be identical), A087803, A000081, A174145 (2nd differences), A005348, A002845, A198683, A187770, A051491.
%K A255170 nonn,easy
%O A255170 1,3
%A A255170 _Vladimir Reshetnikov_, Feb 15 2015
%E A255170 Simpler definition and program in terms of A000081. - _Vladimir Reshetnikov_, Aug 12 2016
%E A255170 Renamed. - _Vladimir Reshetnikov_, Aug 23 2016