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 A255170 a(n) = A087803(n) - n + 1. 5
 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 (list; graph; refs; listen; history; text; internal format)
 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 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... EXAMPLE 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. 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 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 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

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Last modified August 5 22:11 EDT 2020. Contains 336214 sequences. (Running on oeis4.)