1,3

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.

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.

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...

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.

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

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 *)

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

Sequence in context: A267862 A098586 A199812 * A255630 A298535 A110320

Adjacent sequences: A255167 A255168 A255169 * A255171 A255172 A255173

nonn,easy

Vladimir Reshetnikov, Feb 15 2015

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

Renamed. - Vladimir Reshetnikov, Aug 23 2016

approved