OFFSET
1,3
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.
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.
FORMULA
EXAMPLE
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 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vladimir Reshetnikov, Feb 15 2015
EXTENSIONS
Simpler definition and program in terms of A000081. - Vladimir Reshetnikov, Aug 12 2016
Renamed. - Vladimir Reshetnikov, Aug 23 2016
STATUS
approved