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

1, 1, 2, 5, 13, 32, 79, 193, 478, 1196, 3037, 7802, 20287, 53259, 141069, 376449, 1011295, 2732453, 7421128, 20247355, 55469186, 152524366, 420807220, 1164532203, 3231706847, 8991343356, 25075077684, 70082143952, 196268698259, 550695545855, 1547867058852

Offset: 1

Author

Vladimir Reshetnikov, Feb 15 2015

Keywords

• nonn
• easy

Comments

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.

Examples

			a(4) = 1 - 4 + Sum_{k=1..4} A000081(k) = 1 - 4 + 1 + 1 + 2 + 4 = 5.
a(5) = 1 - 5 + Sum_{k=1..5} A000081(k) = 1 - 5 + 1 + 1 + 2 + 4 + 9 = 13.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• Libor Behounek, Ordinal Calculator
• R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis
• MathOverflow, A discussion related to this sequence
• Eric Weisstein's World of Mathematics, Ordinal Number.
• Eric Weisstein's World of Mathematics, Rooted Tree.

Crossrefs

Cf. A199812 (conjectured to be identical), A087803, A000081, A174145 (2nd differences), A005348, A002845, A198683, A187770, A051491.

Programs

Maple

with(numtheory):
t:= proc(n) option remember; `if`(n<2, n, (add(add(
      d*t(d), d=divisors(j))*t(n-j), j=1..n-1))/(n-1))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<2, 0,
      add(b(n-i*j, i-1)*binomial(t(i)+j-1, j), j=0..n/i)))
    end:
a:= proc(n) option remember; `if`(n<3, 1,
      b(n-1$2) +2*a(n-1) -a(n-2))
    end:
seq(a(n), n=1..40);  # Alois P. Heinz, Feb 17 2015

Mathematica

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 *)

Formula

a(n) = 1 - n + Sum_{k=1..n} A000081(k).

Recurrence: a(1) = 1, a(n+1) = a(n) + A000081(n+1) - 1.

Recurrence: a(1) = a(2) = 1, a(n) = A174145(n-1) + 2*a(n-1) - a(n-2).

Asymptotics: a(n) ~ c * d^n / n^(3/2), where c = A187770 / (1 - 1 / A051491) = 0.664861... and d = A051491 = 2.955765...

Extensions

Simpler definition and program in terms of A000081. - Vladimir Reshetnikov, Aug 12 2016

Renamed. - Vladimir Reshetnikov, Aug 23 2016