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%I #40 Aug 23 2016 22:14:09
%S 1,1,2,5,13,32,79,193,478,1196,3037,7802,20287,53259,141069,376449,
%T 1011295,2732453,7421128,20247355,55469186,152524366,420807220,
%U 1164532203,3231706847,8991343356,25075077684,70082143952,196268698259,550695545855,1547867058852
%N a(n) = A087803(n) - n + 1.
%C 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 Alois P. Heinz, <a href="/A255170/b255170.txt">Table of n, a(n) for n = 1..1000</a>
%H Libor Behounek, <a href="http://mujweb.cz/behounek/logic/papers/ordcalc/index.html">Ordinal Calculator</a>
%H R. K. Guy and J. L. Selfridge, <a href="/A003018/a003018.pdf">The nesting and roosting habits of the laddered parenthesis</a>
%H MathOverflow, <a href="http://mathoverflow.net/q/103411/9550">A discussion related to this sequence</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/OrdinalNumber.html">Ordinal Number</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RootedTree.html">Rooted Tree</a>.
%F a(n) = 1 - n + Sum_{k=1..n} A000081(k).
%F Recurrence: a(1) = 1, a(n+1) = a(n) + A000081(n+1) - 1.
%F Recurrence: a(1) = a(2) = 1, a(n) = A174145(n-1) + 2*a(n-1) - a(n-2).
%F Asymptotics: a(n) ~ c * d^n / n^(3/2), where c = A187770 / (1 - 1 / A051491) = 0.664861... and d = A051491 = 2.955765...
%e a(4) = 1 - 4 + Sum_{k=1..4} A000081(k) = 1 - 4 + 1 + 1 + 2 + 4 = 5.
%e a(5) = 1 - 5 + Sum_{k=1..5} A000081(k) = 1 - 5 + 1 + 1 + 2 + 4 + 9 = 13.
%p with(numtheory):
%p t:= proc(n) option remember; `if`(n<2, n, (add(add(
%p d*t(d), d=divisors(j))*t(n-j), j=1..n-1))/(n-1))
%p end:
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<2, 0,
%p add(b(n-i*j, i-1)*binomial(t(i)+j-1, j), j=0..n/i)))
%p end:
%p a:= proc(n) option remember; `if`(n<3, 1,
%p b(n-1$2) +2*a(n-1) -a(n-2))
%p end:
%p seq(a(n), n=1..40); # _Alois P. Heinz_, Feb 17 2015
%t 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 Cf. A199812 (conjectured to be identical), A087803, A000081, A174145 (2nd differences), A005348, A002845, A198683, A187770, A051491.
%K nonn,easy
%O 1,3
%A _Vladimir Reshetnikov_, Feb 15 2015
%E Simpler definition and program in terms of A000081. - _Vladimir Reshetnikov_, Aug 12 2016
%E Renamed. - _Vladimir Reshetnikov_, Aug 23 2016
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